On Designing LyapunovKrasovskii Based AQM for Routers Supporting TCP Flows
Abstract
For the last few years, we assist to a growing interest of designing
AQM (Active Queue Management) using control theory. In this paper,
we focus on the synthesis of an AQM based on the Lyapunov
theory for time delay systems. With the help of a recently developed
LyapunovKrasovskii functional and using a state space
representation of a linearized fluid model of TCP, two
robust AQMs stabilizing the TCP model are constructed. Notice that our results are constructive and the synthesis
problem is reduced to a convex optimization scheme expressed in
terms of linear matrix inequalities (LMIs). Finally, an example
extracted from the literature and simulations via NS simulator [4]
support our study.
Keywords: Active Queue
Management, congestion problem, Linear time delay systems,
LyapunovKrasovskii functional, LMIs.
1 Introduction
Over a past few years, problems have arisen with regard to Quality
of Service (QoS) issues in Internet traffic congestion control
[15], [23]. AQM mechanism, which
supports the endtoend congestion control mechanism of Transmission
Control Protocol (TCP), has been actively studied by many
researchers. AQM controls the queue length of a router by
actively dropping packets. Various
mechanisms have been proposed in the literature such as
Random Early Detection (RED) [6], Random Early Marking
(REM) [1],
Adaptive Virtual Queue (AVQ) [12]
and many others [21]. Their performances have been evaluated [5], [21]
and empirical studies have shown the effectiveness of these algorithms [14]. Then, significant research
has been devoted to the use of control theory to develop more
efficient AQMs. Using dynamical model developed by [17],
some P (Proportional), PI (Proportional Integral)
[10] have been designed as well as robust
control framework issued [20]. Nevertheless, most
of these papers do not take into account the delay and ensure the
stability in closed loop for all delays which could be
conservative in practice.
The study of congestion problem with time delay systems framework is not new and has been succesfully
exploited. In [16], [18], using LyapunovKrasovskii theory, the global stability analysis of
the non linear model of TCP is performed. In [11], a delay dependent state feedback controller is provided by
compensation of the delay with a memory feedback control. This latter methodology is interesting in theory but hardly suitable
in practice.
Based on a recently developed Lyapunov
functional for time delay systems, two AQMs stabilizing
the TCP model are constructed. The first one is called IODAQM (Independent Of
Delay) and it deals with the robust control of TCP for all delays
in the loop. The second one, DDAQM (Delay Dependent) is
devoted to the control of the TCP dynamics when an upperbound of
the delay is known. In order to consider a more realistic case,
extension to the robust case, where the delay is uncertain is considered using quadratic
stabilization framework.
The paper is organized as
follows. The second part presents the uncertain mathematical model
of a network supporting TCP. Section III is dedicated to the
design of two AQMs ensuring the robust stabilization of TCP. Section IV presents application
of the exposed theory and the simulation results
using NS2. Finally, section V concludes the paper.
Notations: For
two symmetric matrices, and , () means that
is (semi) positive definite. denotes the transpose of
. and denote respectively the
identity matrix of size and null matrix of size . If
the context allows it, the dimensions of these matrices are often
omitted. For a given matrix , stands for .
2 Problem statement
2.1 The linearized fluidflow model of TCP
The fluid flow model of TCP considered here was
introduced in [17], [10]. Based on this system, we
will construct two AQM, which take into account
delays inherent to networks.
Given the network
parameters: number of TCP sessions, link capacity and propagation
delay (, and respectively), we define the set of
operating points by and :
(1) 
where is the congestion window, is the queue length at the congested router and is the Round Trip Time (RTT) which represents the delay in TCP dynamics. denotes the value of the variable at the equilibrium point.
Assuming and as constants,
the dynamic model of TCP can be approximated, around an
equilibrium point, by the linear time delay system [17]:
(2) 
where , and are the state variables and input perturbations
around the operating point. The model (2) is valid only
if the variations of these new variables are kept enough
small.
The input of our model (2) corresponds to the drop probability of a packet.
This probability is fixed by the AQM. This latter has for objective to regulate the queue size at the router.
For synthesis problem (see section 3), we
consider a state feedback. So that, the queue
management strategy of the drop probability will be expressed as
(3) 
Remark 1
i) It is possible to design a state feedback as it corresponds to a PD (Proportional Derivative) control law [11]. Furthermore, although is not measured, one can estimate , the aggregate flow at the link [11], [15].
ii) The main difficulty in all representations of TCP behavior is
the exact estimation of network parameters (and not the state
feedback control law). Two techniques
are used:
Active measurements [13], [19] consist in generating probe traffic in the network, and
then observing the impact of network components and protocols on
traffic: loss rate, delays, RTT, capacity… Therefore, as active
measurement tools generate traffic in the network (intrusiveness), one of their
major drawbacks is related to the disturbance introduced by the
probe traffic which can make the network QoS change, and thus
provide erroneous measures [13]. Sometimes, active probing
traffic can be seen as denial of service attacks (DoS), scanning, or
something else but in any case as hacker acts. Probe traffic is then
discarded, and its source can be blacklisted.
Passive measurements refer to the
process of measuring a network, without generating or modifying any
traffic on the network. Passive monitoring is done with the capture
of traffic and estimate off line networks parameters: It’s still to
be non intrusive (good estimation of parameters) but not reactive.
The passive evaluation relies on DAG system cards [3] that
represent references for such kind of measurements.
Passive and active measurements is still a growing interest because exact estimation of networks parameters still difficult since the heterogeneity of autonomous systems [13]. A future idea (early introduced in this paper) for this problem is to consider uncertainties for parameters: This solution allows to use robust control theory in sense of polytops.
2.2 Time delay system approach
In this paper, we choose to model the dynamics of the queue and the
congestion window as a timedelay system. Indeed, the delay is an
intrinsic phenomenon in networks. Taking into account this
characteristic, we expect to reflect as much as possible the TCP
behavior, providing more relevant analysis and synthesis
methods.
The linearized TCP fluid model (2)
can be rewritten as the following time delay system:
(4) 
with
(5) 
where is the state vector and the input.
is the initial condition.
There are mainly
three methods to study time delay system stability: analysis of the
characteristic roots, robust approach and Lyapunov theory. The
latter will be considered because it is an effective and practical
method which provides LMI (Linear Matrix Inequalities
[2]) criteria. To analyze and control the system
(4), the LyapunovKrasovskii approach [9] is
used which is an extension of the traditional Lyapunov
theory.
In the literature, few articles using time delay
systems approach to model TCP dynamic already appeared. In
[24], a delay dependent robust stability condition was
proposed and the design of a state feedback was derived. However,
the criterion used is quite obsolete and thus conservative. Then,
other papers design control laws based on predictor [11].
The predictive approach is an interesting method theoretically but
not in practice, moreover the delay has to be known exactly.
[16] and [18] use time
delay system approach too and propose global stability analysis of the linear model.
However synthesis is not considered.
In this paper, we aim at providing
methods to control system
(4) with different objectives: giving conditions for
the nominal or robust stabilization for IOD and DD cases.
2.3 Polytopic uncertain model
The state space representation shows that matrices ,
and depend on network parameters. Especially, it depends on the
RTT , a significant parameter, which is difficult to
estimate in practice. For a more rigorous study, it could be
interesting to take into account some uncertainty on the delay
.
Let then rewrite system (4) as
following
(6) 
With the polytopic approach, the idea is to insure the stability for a set of systems. Let suppose that , then matrices , and belong to a certain set
and we aim at looking for an AQM (expressed in term of state
feedback) which stabilizes system (6) for all matrices
belonging to . However, the parameter doesn’t appear
linearly in the matrices , and . So that, the set
defined by the uncertainty is non
convex.
A common idea in robust control theory is to look for
a polytopic set which includes the set . Using
convexity property, it is much more easy to test the stability in
closed loop for the overall polytop. If the stability of
is proved, then the stability of is
insured.
In order to create the polytop , we pose
, and
. Since there are three uncertain parameters, the
polytop will have vertices. For a bounded value ,
the new uncertain parameters , are
bounded. So, the matrices of the uncertain system (6)
are defined as
(7) 
The set is contained in the set ,
where the are the vertices of .
3 Stabilization using timedelay system approach
In the previous section, an uncertain model of the TCP/AQM dynamic has been designed. This section is devoted to the construction of robust AQM stabilizing a such model. The first approach proposes the construction of an independent of delay (IOD) controller using convex optimisation schemes (LMI). In a second part, we describe a delay dependent (DD) method which takes into account the size of the delay. Using an information on the delay, we expect a reduction of conservatism and then an improvment of results.
3.1 Independent of delay AQM design
The idea is to insure the stability in closed loop for all delays as it has been proposed in [10] using frequential arguments and traditionnal control tools. Here, we propose to use the following wellknown LyapunovKrasovskii functionnal:
(8) 
where the matrices and are symmetric and positive definite. The choice of this LyapunovKrasovskii functional implies the following proposition.
Proposition 1
Now, we construct the following memoryless state feedback
(10) 
to control system (4) ( is a constant matrix gain). This controller corresponds to our AQM.
Applying (10) to (4), we get the
closedloop system
(11) 
with .
Then, the following synthesis criterion can be easily derived from (9) and (11).
Proposition 2
This latter proposition provides an IODAQM, , which stabilize (4) for all delays .
3.2 Delay dependent AQM design
In this subsection, our goal is to design a controller which takes into account the upperbound of the delay. The delay dependent case starts from a system stable without delays and looks for the maximal delay that preserves stability.
Generally, all methods involve a Lyapunov functional,
and more or less tight techniques to bound some cross terms and to
transform system [9]. These choices of specific
Lyapunov functionals and overbounding techniques are the origin of
conservatism. In the present paper, we choose a recent
LyapunovKrasovskii functional (13)
[7]:
(13) 
where is a positive definite matrix, and are two positive definite matrices. is an integer corresponding to the discretization step. Using this functional, we propose the following.
Proposition 3
If there exist symmetric positive definite matrices , , , a matrix , a scalar , an integer and a matrix such that
(14) 
where
(15) 
and
then, system (4) can be stabilized for all by the control law .
Proof 1
Remark 2

There exists another equivalent form of this LMI in term of analysis (i.e. with ) provided in [7] and based on robust control tools.
Nevertheless, applying a state feedback (10),
we have with the controller gain appearing
as a decision variable. Then, the condition becomes a BMI. That’s the reason why in this paper, we propose a relaxation algorithm. The algorithm principle
consists to alternate analysis and synthesis steps.
First
let define the synthesis LMI:
(19) 
where and
is the slack variable which has been
fixed.
By the same way, we define the analysis LMI:
(20) 
where is fixed. Then, we propose the following
algorithm.
Algorithm:

We solve the synthesis optimization
A matrix gain called is derived.

We solve the analysis optimization with .
The new slack variable is derived .
We test if .

if true, there is no improvement on the maximal size of the allowable delay: end of the algorithm.

if false, the process is reiterated to the step (1) with a new slack variable and upperbound of the delay.
Remark 3
At the test step, one always has . Consequently, throughout the progression of the algorithm the upperbound can not regress.
Notes that the main problem, which is common in relaxation methods, remains the initialization of slack variables.
4 Application to TCP/AQM dynamics and validation through NS2
In this section, we are going first to consider the nominal system in order to expose the control principle. Then, we will extend our methods to the robust case. For a realistic case, it is essential to insure stability in spite of the delay uncertainty.
4.1 Numerical example
As a widely adopted numerical illustration extracted from [10], consider the case when packets, second and packets/s (corresponds to a Mb/s link with average packet size bytes). Then, for a load of TCP sessions, we have packets, , seconds. We obtain the following open loop system
(21) 
Matrix is Hurwitz and applying IOD proposition 1, we observe that the LMI (9) is feasible. So, we conclude (21) is IOD stable and system (21) is stable for all . However, in order to avoid congestion and to regulate the queue size at a desired level in spite of uncertainty on delay, an AQM has to be implanted.
4.2 Iod/Dd Synthesis
Independent of delay method
In the IOD case, for nominal system only, it turns out that the delayed term, which can be viewed as a disturbance, can be eliminated choosing and as:
(22) 
Thus if is Hurwitz and
for a state feedback gain defined as (22), then the
system (4) is IOD stable. Since an IOD stabilizing gain K can always be found (in nominal case), this method provides a systematic technique for the algorithm initialization (for DD synthesis).
Concerning the
robustness issue, let consider that is uncertain such
that . This system will be
stable if the polytop (see section 2.3) is
stabilized. Using the quadratic stability framework [2], we propose the
following result.
Proposition 4
Taking numerical example (21), nominal
value is seconds. The objective is to stabilize the uncertain system (6)
for given bounds ( and ), and to
maximize the stability domain.
Applying IOD LMI condition of the proposition
4, we obtain the results of table I.
Gain  

We can observe, in that case, decreasing the lowerbound is more restrictive than increasing the upperbound for the optimization problem. Nevertheless, because of the unavoidable delay in networks (like propagation delays), it is useless to look for a very small lowerbound.
Delaydependent method
Using the relaxation algorithm previously exposed and IOD gain (22) for the initialization, we get the following results of the table II for the robust delay dependent case where a common LyapunovKrasovskii functional is found for each vertice of the polytop. Compared to IOD results, we improve sligthly the set of admissible delays.
Gain  

Remark 4

If , then system (6) is just stable for since is the RTT and corresponds to the delay.

As expected, we obtain better results for , since is larger.
4.3 Simulations
We aim at proving the effectiveness of our method using NS2
[4], a network simulator widely used in the communication
community. Taking values from the previous numerical example, we
apply the new AQM based on a state feedback (i.e a simple constant
matrix gain ). The target queue length is packets
while buffer size is . The average packet length is
bytes. The default transport protocol is TCPNew Reno without ECN
marking.
For the convenience of comparison, we adopt the
same values and network configuration than [10] who design
a PI controller (ProportionalIntegral). This PI is configured
as follow, the coefficients and are fixed at and
respectively, the sampling frequency is
Hz.
In the figure 1, we apply the gain
from the table II which ensures DD robust stability. We compare our result with PI AQM
provided by [10]. It appears that our control allows a faster response as
well as a smaller overshoot.
Simulations of perturbed system is reported in figures 2 and 3. In figure 2, we have increased the propagation delay by ms. Even if the system converges to a different reference point (slightly lower), the queue size is stable and quickly regulated.
For more important pertubations (on the delay or number of sessions ), the system in closedloop is still stable but the steady state changes since we converge to a new equilibrium point. In figure 3, a gain is calculated from DD robust stabilization with an external perturbation. The scenario is composed as follows: 7 additive sources (UDP protocol) send 1000 bytes packet length with a 1Mbytes/s throughput between and . With the DD robust controller, the response is perturbed. The closedloop system converges to the same reference, the queue size is stable and quickly regulated when the perturbation disappeared.
5 Conclusion
In this preliminary work, we have proposed the construction of robust AQMs for the congestion problem in communications networks. The developed AQMs have been established by using Lyapunov theory extended to delay systems and semi definite programming to solve the Linear Matrix Inequalities. Note that the proposed methods have been extended to the robust case where the delay in the loop is unknown. Finally, the AQMs have been validated using NS simulator.
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