Analytical Design of First-Order Controllers for The Tcp/Aqm Systems with Time Delay

7/31/2019 Analytical Design of First-Order Controllers for The Tcp/Aqm Systems with Time Delay

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International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012

DOI:10.5121/ijitca.2012.2303 27

ANALYTICAL DESIGN OF FIRST-ORDER

CONTROLLERS FOR THETCP/AQM SYSTEMS WITH

TIME DELAY

Sana Testouri1

and Karim Saadaoui1

and Mohamed Benrejeb1

1Department of Electric Engineering, National School of Engineering of Tunis

[email protected]

ABSTRACT

In this paper, the AQM controller of a first-order controllers type is proposed. The model of TCP/AQM is

described by a second-order system with time delay. An analytical approach to analyze the stability ofTCP/AQM Networks is used, based on the D-decomposition method and lemma Kharitonov for quasi-

polynomial. The proposed method for design first-order controller is verified and compared with other

existing AQM schemes, using NS-2 simulator.

KEYWORDS

Stability, Time delay, First-Order Controller, Active Queue Management

1. INTRODUCTION

The congestion-control mechanism becomes indispensable in an over-charged network. TCP(Transmission Control Protocol) has been the basis of congestion control. It adopts the end-to-end

window-based flow control to avoid congestion [1]. Recently, a growing interest in designingAQM (Active Queue Management) has been proved an efficient approach to enhance congestion

control, indeed, a significant research has been devoted to the use of control theory to developmore efficient AQM [6,7,8]. The goal of AQM is to maintain shorter queuing delay and higher

throughput by dropping packets at intermediate nodes. It has therefore attracted attention in the

research for transmission control protocol (TCP) of end-to-end congestion control. Random earlydetection (RED) [2] is the first well known AQM algorithm, which aims to drop packets with a

certain probability as a function of the average queue size. Furthermore, it is difficult to obtainadequate values of RED parameters which provide satisfactory performance in terms of overallquality of service (QoS). Central to our approach is utilizing the reformulated AQM schemes as a

controller for congestion control. Consequently, feedback control principles appear to be anappropriate tool in the analysis and design of AQM strategies. Using dynamical model

TCP/AQM, some P (Proportional), PI (Proportional Integral) have been designed [3], thisnonlinear model is linearized at an operating point to address the feedback control nature.

Proportional (P) and proportional-integral (PI) AQM controllers were introduced by comparison

to RED. A structural nonlinear component of RED was considered, and the describing functionapproach was applied to obtain a stability criterion for RED and PI controller was proposed,

which adaptively adjusted the controller parameters based on network parameters estimation [4],[5]. In order to stabilize the network traffic system and achieve the desired QoS, [3],[6],[7] paid

much attention to the setting of controller parameters. Our objective is to obtain the stabilizing
mailto:[email protected]:[email protected]


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regions of a first-order controller in order to choose parameters which stabilize the systemmodeled by a second-order plant with time delay. In earlier works, with a P type AQM controller,

in the case of delay-free marking, the systems equilibrium point is stable for all proportionalgains. In a more realistic case of delayed feedback, there exists a boundary for proportional gain

to guarantee the closed-loop AQM system stability [8]. When the AQM controller is of PI type,the stabilizing boundary of proportional gain has been given by using a parameter space approach[7]. Our approach consists of determining the stabilizing regions in the parameter space of the

first-order controller, then choosing controllers parameters within these regions. Finally, we

illustrate the proposed methodology with an example and simulations experiments implemented

by both Matlab and Network Simulator NS-2 for validate our analysis. In this paper, the AQMcontroller is of first-order controller type is presented. A parametric approach for stabilizingclosed-loop for AQM system is proposed based on the D-decomposition method and lemma

Kharitonov for quasi-polynomial. Section 2 introduces the TCP/AQM dynamic model. Next, thestabilizing regions in the parameter space of the controller are determined in section 3. Finally, in

section 4, simulation results both with Matlab and NS-2 are given.

.2. DYNAMIC MODEL OF AN AQM ROUTER

2.1. Fluid-Flow Model of TCP Behaviour

The dynamic model of TCP flows is developed by using a fluid flow model without consideringslow start and timeout mechanisms [4] .Based on this system, a type of AQM is constructed,

which takes into account delays into the network.

This model is described by the following non-linear differential equations

( )

( )( )

( )( )

( )

.1 ( ) ( ( ) )

( ( ) ) ,( ) 2 ( ( ) )

,

( ).

( )

W tW t W t R t

p t R tR t R t R t

W tq t N t C

R t

q tR t T

pC t

=

=

= +

(1)

where ( )W t and ( )q t denote the time-derivatives of ( )W t and ( )q t , respectively. ( )W t denotes theTCP window size, ( )q t denotes the queue length in the router.

( )p t denotes the probability packet marking/dropping [ ]( )( ) 0,1p t . ( )R t denotes the round-trip time,

( )C t denotes the link capacity. Tp

denotes the propagation delay. ( )N t denotes the load factor

(number of TCP sessions).The first differential equation in (1) describes the TCP window control

dynamic and the second equation models the bottleneck queue length. The queue length and

window size are positive, bounded quantities, i.e., 0,q q , [ ]0,W W window size, respectively.Also, the marketing probability p takes value only in [ ]0 ,1 .

In this model, the congestion window ( )W t increase linearly if no packet loss is detected;

otherwise it halves


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2.2. Linearization

Although an AQM router is a non-linear system, in order to analyze certain types of propertiesand design controllers we need a linear model which is presented in this sub-section. To linearize

(1), we first assume that the number of TCP sessions and link capacity are constant,

i.e., ( )N t N

, ( )C t C

Taking ( ),W q as the state and p as input, the operating point ( ), ,0 0 0W q p is then defined by 0W = and

0q = so that

2

0

.

0

.0

0

0

0

0 2,

0 ,

p

W W p

R Cq W

N

qR T

C

= =

= =

= +(2)

We linearize (1) about the operating point to obtain

2

00 02 2

0

0 0

( ) ( ( ) ( )) ( ),2

1( ) ( ) ( ).

R CNW t W t W t R p t R

R C N

Nq t W t q t

R R

= +

=

(3)

where0

W W W = ,0

q q q = ,0

p p p = represent the perturbed variables around the operating

point.

For typical network conditions [4],N 1 1

=


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3. STABILIZING FIRST-ORDER CONTROLLERS

We consider the closed- loop AQM system with ( )C s the transfer function of the controller,

and ( )Gs the transfer function of the plant dynamic. This model presents the dynamics of the queue

and the congestion window as a time delay system.

Indeed, taking into account this characteristic, we expect to reflect the TCP/AQM behaviour incontrol congestion.

.

Figure 1. Block diagram of the AQM system

( ) ( )

( )

0( )B R s

G s e

Q S

G s G sT C P q u e u e

== (6)

where2

2

CB

N= , ( )

2 1( ) ( )

2 00

NQ s s s

RR C= + + .

As the network paramet { }, , 0N C R are positive, where 00R > is the time delay, and ( )C s is the first

order controller having the form

(7)

The closed-loop AQM system is a second-order system with time delay, whose characteristicequation is

( ) ( )1 0C s G s+ = (8)

which leads to the following characteristic quasi-polynomial

(9)

Multiplying both sides of (9) by 0Rs

e yields

(10)

As0sR

e

does not have any finite zeros [10], the zeros of*

( )sV are identical to those of ( ).sV Thecharacteristic quasi-polynomial * ( )sV of the closed-loop AQM system is stable if and only the

zeros of ( ).sV are in open left half plane (LHP). Then, ( ).sV is defined as Hurwitz or stable.

Determining stabilizing controller parameters ( )1 2 3, , will be done in the next section.

2 3( )1

sC s

s

+=

+

0( ) ( ) ( ) ( )1 2 3

R ss s Q s e B s = + + +V

* 0( ) ( ) ( ) ( )1 2 3

sRs s Q s B s e

= + + +V


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3.1. Computing the admissible values of 1

First, the admissible values of1

are calculated, the following Lemma, gives a condition for the

stability of ( )sV , where ( )sV denotes the derivative of ( ).sV

Lemma 1. [12] Consider the quasi-polynomial

(11)

such that


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where ( )R and ( )Iare the real and the imaginary part of ( )Q j

Three cases will be investigated:

Case1. Setting 0= , this leads to the following equation:

( )3 11

0RB

=(15)

Case2. For 0> , the following pair of( ),2 3 can be calculated for each fixed value of 1 :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 1 1

3 1 1

1( ) sin cos( )

1( ) cos sin( )

R II L R L

B

I R L R I LB

= +

= + + (16)

By sweeping over all values 0 > , the ( ),2 3 plane can be partitioned into rootinvariant regions,

therefore stabilizing regions can be determined. Repeating the same steps for the admissible

values of 1leads to the determination of all stabilizing ( ), ,1 2 3

values.

3.3. Control Action

AQM controller marks packets with a probability. The marking probability is calculated

according to the first-order controller and it is a function of the difference between the

instantaneous queue length and the desired queue length to which we want to regulate. q is given

by0

q q q = , and we assume p =0 , which makes p p = . In practice the sampled queue system

needs the discrete form of controller, after substituting a series of sampling time kTfor continuestime tin equation (7), we obtain the increment expression of the form:

( ) ( ) ( )( ) ( )( )1 1 *q qp kT a kT b k T p k T = + (17)

For implement in ns2, the first order controller we have need convert the increment expression bypseudo code with a sampling frequency of 160 Hz [8].

1

0 0* ( ) ( ) *

o l d o l d

o l d

o l d

T

p a q q b q q p

p p

q q

e

= +

=

=

= (18)

4. SIMULATION

4.1. Simulation in Matlab

For determining stabilizing controllers for the system (6), we consider the network parameters

N = 60,C = 3750packets / s and0

0.25R s= and we obtain (19)

117187.5 0.25( )

( 0.512 )( 0.4)

sP s e

s s

=

+ + (19)


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Applying the procedure given in the previous section, the admissible range of 1

is found tobe 2

1 > .

Now, we choose the admissible values within therange = 0 . 5 e - 5 , 0 . 6 e - 5 ; 0 . 7 e - 5 , 0 . 8 e - 5 , 0 . 9 e - 5 , 1 e - 5

1The stabilizing region in the plane

ofthe remaining of the two parameters ( ),2 3 derived from equations (15) and (16) is determined.Figure. 2 give the set stabilizing first-order controller. Setting =0.6e-51 , we have two groups ofcontroller parameters. ( )10 20 30, , locates inside the stabilizing region, ( )10 21 31, , locates outsidethe stabilizing region.

3

4

5

6

7

x 10-6

-1

0

1

2

x 10-4

-1

0

1

2

x 10-4

12

3

Unstable

Stable

Figure 2. Stabilizing regions of controller parameters.

Figure.3 represents the step response of the closed loop system with

( )10 20 30-4-5 -4

, , (0.6 )= and ( ) 5 4 410 21 31- - -

, , ( 0 .6=

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

2.5Step Response

Time (sec)

Amp

litude

10

,20

,30

10

,21

,31

Figure 3. Step responses of the closed loop system.

We find that ( )1 0 2 0 3 0, , leads the output convergence, and ( )10 21 31, , leads the output

divergence.

Now, we turn our attention to analyzing the relations between stabilizing boundary of the gain in

first-order controller and the network parameters [6, 7].For [ ]N 50; 300 , according to the stabilitycriterion given in the previous section, we check the stabilizing boundary of

1gain by (13) and

we plot of the curve1( shown in Figure 4.(a)


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50 100 150 200 250 300-3.2

-3.1

-3

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2

N

alpha

1

1500 2000 2500 3000 3500 4000 4500 5000-2.7

-2.65

-2.6

-2.55

-2.5

-2.45

-2.4

-2.35

-2.3

-2.25

-2.2

C

alpha

1

(a) Stabilizing region of1

for differentN. (b) Stabilizing region of 1for different C.

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

R0

alpha

1

(c) Stabilizing region of1

for different0

R .

Figure 4. Stabilizing regions for different network parameters.

(The stabilizing region is above the curve).

Also, we plot of the curve ( , )1C for 0N= 60,R =0.25sand ranges [C1500;500 shown the stabilizing region

of1

in Figure 4. (b) and for N= 60,C =3750packets / sand [ ]0R 1500;5000 .

From these two curves, we find that stabilizing boundary of increases with N and decreases with

C and0R ,respectively

4.2. Simulation in NS-2

To verify the stabilizing parameters in network environment, we conducted non linear simulationby NS, using the network topology depicted in Figure5.

Figure 5. Simulation of network topology.


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We introduced 60 TCP flows and the simulation time is 80 s. ( )1,...,S i ni

= are TCP senders with

average packet size 1000 Bytes. Sd

is a FTP sender which has 10 Mbps capacity and 20 ms

propagation delay, the traffic scenario. The only bottleneck link lies between RouterA

RandRB

,

which has 15Mbps capacity and 5ms propagation delay. Router1

R

uses the First-order controller,others use the Drop Tail. The sampling is 160H z . The buffer size is 800 packets and the desired

queue length is 200 packets. As one of the controller parameters inside the stabilizing region,

( )1 2 3, , regulates the queue length to the desired 200 packets. We will compare the dynamicresponse of the firstorder controller with that Hollots PI controller in [5]. First, we plot thequeue lengths for different first-order controller parameters, inside and outside the stabilizing

region.

Figure 6 shows the instantaneous queue length, ( )10 20 30, , regulates the queue length to the desired200 packets pictured in Figure 6. (a).

Figure (b) shows significant oscillation in the case ( )10 21 31, , outside the region, when it isunstable.

The performance of first-order controller is compared by PI controller presented in Figure 6. (c).

It clear that the queue length with the first-order controller is fast, more stable and overshoot isnegligible than Hollots PI controller, the PI controller takes long time to settle the queue lengtharound target.

(a)( )10 20 30, , inside the stabilizing region. (b) ( )10 21 31, , outside the stabilizing region.


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(c) Intantaneous queue length for PI controller.

Figure 6. Instantaneous queue lengths for first-order controller for different parameters and PI controller.

For validate our analysis, the curves of the dropping probability are plotted in Figure 7.

The regulating time of PI controller is almost eight times larger than that of first order controller.

Figure 7. Instantaneous drop probabilities for first-order controller and PI controller.

5. CONCLUSION

This paper discusses the stability characteristic of TCP/AQM networks using control theory. The

stabilizing regions of a first-order controller are given. Simulation experiments conducted byMatlab and NS have validated our criterion. In this paper, we have just determining stabilizing a

first-order controller for TCP/AQM system with time delay. therefore, in order to achieve better

performance criterion (no over-shoot, minimal rise time, Steady state error = 0), we will proposein our futur work, a method for determining the optimal parameters of first -order controller.


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REFERENCES

[1] Van Jacobson, (1988) Congestion avoidance and control Proceeding of SIGCOMM88, ACM.

[2] Sally Floyd and Van Jacobson, (1993) Random early detection gateways for congestion

Avoidance, IEEE/ACM Transactions on Networking, Vol. 1, pp 397-413.

[3] C V Hollot, Vishal Misra, Don Towsley and Wei-Bo Gong, (2002) "Analysis and design ofcontrollers for AQM routers sup porting TCP flows". IEEE Transactions on Automatic Con trol,

vol.47, pp 945-959.

[4] C V Hollot, Vishal Misra, Don Towsley and Wei-Bo Gong, (2002) A Control Theoretic Analysis of

RED, Proceedings of IEEE Infocom .

[5] C V Hollot, Vishal Misra, Don Towsley and Wei-Bo Gong, (2001) On designing improved

controllers for AQM routers supporting TCP flows, Proceedings of IEEE INFOCOM, pp 1726 -1734

.

[6] Ge Long, Fang Bin, Sun Jin-Sheng and Wang Zhi-Quan, (2010) Novel graphical approach to analyse

the stability of TCP/AQM networks, Acta Automatica Sinica, Vol.36 , No. 2, pp 314 -321.

[7] Wang De-Jin, (2007) Stability analysis of PI TCP/AQM networks: a parameter space approach ,

Acta Automatica Sinica, Vol.33, pp 756-760.

[8] C. V. Hollot and Yossi Chait, (2001) "Nonlinear stability analysis for a class of TCP/AQM

networks", Orlando, USA: IEEE, pp 2309-2314.

[9] Online Available: NS-2 Simulator, available at http://www.isi.edu/nsnam/ns/

[10] Guillermo J Silva, Aniruddha Datta and S P Bhattacharyya, (2001) PI stabilization of firstorder

systems with time delay, Automatica, Vol. 37, pp 2025-203.

[11] Karim Saadaoui, Sana Testouri and Mohamed Benrejeb, (2010) Robust stabilizing first -order

controllers for a class of time delay systems ", ISA Transactions Vol. 49, pp. 277-282.

[12] Kharitonov. V. L. Niculescu S, Moreno J et Michiels W. (2005)"Static output stabilization: Necessary

conditions for multiple delay controllers, IEEE Trans Automat. Control.

[13] Gryazina EN, Polyak BT (2006) "Stability regions in the parameter space: D-decomposition

revisited", Automatica;42, pp 13-26.
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Analytical Design of First-Order Controllers for The Tcp/Aqm Systems with Time Delay

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