-
Accepted Manuscript
Design of adaptive backstepping congestion controller for TCP
networkswith UDP flows based on minimax
Zanhua Li, Yang Liu, Yuanwei Jing
PII: S0019-0578(19)30214-9DOI:
https://doi.org/10.1016/j.isatra.2019.05.005Reference: ISATRA
3200
To appear in: ISA Transactions
Received date : 29 July 2018Revised date : 28 April 2019Accepted
date : 3 May 2019
Please cite this article as: Z. Li, Y. Liu and Y. Jing, Design
of adaptive backstepping congestioncontroller for TCP networks with
UDP flows based on minimax. ISA Transactions
(2019),https://doi.org/10.1016/j.isatra.2019.05.005
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https://doi.org/10.1016/j.isatra.2019.05.005
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*Corresponding Author (e-mail: [email protected],
[email protected] )
Design of adaptive backstepping congestion controller for
TCP
networks with UDP flows based on minimax
Zanhua Li1,2*, Yang Liu1, Yuanwei Jing1*
1.College of Information Science and Engineering, Northeastern
University, Shenyang, 110819, China
2.School of science, Shenyang Ligong University, Shenyang,
110168, China
Abstract: The congestion control problem of TCP network systems
with user datagram protocol (UDP) flows is
investigated in this paper. A nonlinear TCP network model with
strict-feedback structure is first established. The
unknown UDP flow is regarded as the external disturbance, and
the maximum UDP flow is calculated by using the
minimax approach. And then, a congestion control algorithm is
proposed by using the adaptive backstepping
approach. Meanwhile, the adaptive law is employed to estimate
the unknown link capacity. The design of the
adaptive law is to introduce a parameter mapping mechanism to
limit the parameter identification range to a
specified interval, thereby improving the estimation efficiency
of the parameters. Furthermore, a state-feedback
congestion controller is presented to make sure that the output
of the system tracks the desired queue. The
simulation results show the superiority and feasibility of the
proposed method.
Keywords: TCP network; congestion control; minimax; adaptive
backstepping technique.
1 Introduction
Recently, the active queue management (AQM) has been a very
active research area in the internet community
[1]-[3]. Though there exist a lot of achievements, the
congestion control problem based on AQM is still an open
field. In 2000, Misra et al. [4] established a nonlinear
differential equation model for router queues of TCP
networks based on the fluid-flow theory. Hollot et al. gave a
reasonable linearization for the nonlinear dynamic
model of TCP network in [5]. Authors of [6]-[7] proposed an AQM
algorithm based on the proportional integral
(PI) and proportional differential (PD), respectively. Authors
of [8] investigated a robust fractional-order PID
controller for time-varying parameters of the network system. So
far, a number of results have been obtained to
solve network congestion problems. All the results require the
system to be linear. The congestion control
mechanism of the TCP protocol is considerably complex, in which
many nonlinear distortion factors appear.
Hence, an AQM algorithm was proposed based on the nonlinear
network model in [9]. Authors of [10]-[12]
designed an AQM algorithm by utilizing the fuzzy
variable-structure control and neural-networks method,
respectively. Particle swarm optimization (PSO) was applied to
obtain the output weight of radial basis function
(RBF) neural networks, and then an AQM controller was achieved
in [13]. Authors of [14]-[15] considered a
situation where the external disturbances existed in network
systems and an AQM algorithm was presented.
*Title page showing Author Details
-
It is well known that the backstepping technique is one of the
main control methods for nonlinear systems. In
recent years, this method has been widely used in many fields
and a great deal of results have also been obtained
[16]-[20]. In [16], the backstepping approach was adopted to
solve the problem of global uniform asymptotic
stability for nonlinear systems with an arbitrarily large delay
of the input. Authors of [17] investigated the robust
control problem by the backstepping method for time-delay
nonlinear systems with triangular structure, in which
the system with time-delay was considered. In [18], a robust
stabilization issue was studied by using the adaptive
robust backstepping control method for structure uncertain
nonlinear systems in the presence of structured
uncertainties, external disturbances, and unknown time-varying
virtual control coefficients. In [19], a finite-time
command filtered backstepping approach was proposed for
high-order nonlinear systems. Authors of [20]
designed an optimized backstepping control technique to solve
the optimized solutions for the high-order strict-
feedback systems. In addition, the backstepping technique has
been applied to network systems [21]-[23]. In [21],
the nonlinear output-feedback control algorithm was obtained
based on the comparison lemma and backstepping
technique, and the range of parameters was also proposed.
Authors of [22] designed an AQM controller with only
one output (queue size) measurement, the control law was
developed by applying an observer-based backstepping
design technique. In [23], the prescribed performance,
backstepping technique, adaptive control and H∞ control
were combined to design a congestion controller. Although the
research of congestion control strategy has made
great progress, there are still some problems that have not been
fully solved. One of the most important issues is to
deal with disturbance and the uncertainty in the network
system.
In the existing achievements [24]-[25], on one hand, the
disturbance is not considered in the system; on the
other hand, an assumption that the disturbance has an upper
bounded is required and the upper bound is used to
replace the disturbance. The above two situations can cause
certain limitations and conservatism, hence, the idea
of minimax method is an effective scheme to handle this problem.
Compared with the existing ways, the minimax
control method aims at calculating the disturbance with the
worst case, and then discusses the design of controller
to provide better disturbance rejection characteristics. It is
well known that the minimax method is to construct a
controller to stabilize the system under the case of the
worst-case uncertainty. In recent years, the minimax
method has been applied to many systems [26]-[32]. Authors of
[26] proposed minimax guaranteed cost-control
for uncertain nonlinear systems. Authors of [27] focused on a
minimax optimal problem for stochastic systems,
and the corresponding minimax controller was designed with the
worst-case uncertainty. A new minimax control
method was proposed by using universal learning networks in
[28]. Authors in [29] used a linear minimax
observer to study the problem of sliding-mode control design for
linear systems with incomplete and noisy
measurements of the output and exogenous disturbances. Authors
of [30] dealt with the robust and non-fragile
minimax control problem for a T-S model including the parametric
uncertainty terms of the nonlinear systems. A
robust minimax linear quadratic gaussian (LQG) controller was
designed based on an uncertain system model,
which was constructed by measuring the plant variations and
modeling the error between the measured and
modelled frequency-responses in [31]. A stochastic minimax
optimal time-delay state feedback control strategy
for uncertain quasi-integrable Hamiltonian systems was proposed
in [32]. The minimax method was also applied
recently to AQM computer network, which was a kind of typical
nonlinear systems with the nonlinear structure
and time-varying parameters. Authors of [33]-[34] proposed an
AQM controller for linearized congestion router
network systems in the presence of unknown time-varying link
number and disturbances based on the idea of
-
minimax method.
On the other hand, the adaptive technique was an efficient
method for the uncertain systems [35]. Therefore, in
order to handle unknown network situations, some researchers
adopted adaptive control methods. Authors of [36]
designed a controller, which can adapt to unknown or slowly
varying parameters by using the feedback
linearization and the backstepping technique. An adaptive
generalized minimum variance congestion controller
was proposed in [37] based on active queue management (AQM)
strategy. Sun et al. [38] gave an adaptive
proportional-integral controller, which was robust with respect
to non-responsive flows. In order to overcome the
drawback of the Generalized Minimum Variance method, authors of
[39] proposed a wavelet neural network
control method for AQM in an end-to-end TCP network, which was
trained by adaptive learning rates. A few
results have also designed some adaptive congestion controllers
to deal with network congestion for TCP/AQM
system with unknown parameters [40]-[42].
Inspired by the above discussions, the contributions of this
work are summarized as follows: (1) This paper is
first to focus on the congestion control problem for TCP/AQM
network by combining the adaptive backstepping
method and minimax technique. (2) The existence of the unknown
link bandwidth C and external disturbance
UDP flows makes network design difficult. However, the adaptive
control and minimax approaches can be used
to address the two cases mentioned above. Therefore, the two
control methods are combined to study the
congestion control problem. (3) The effectiveness and
superiority of the proposed approach is verified by
comparing with the existing results. The rest of this paper is
arranged as follows. Section 2 illustrates the model of
TCP network with unresponsive UDP flows. The design of the
adaptive backstepping controller based on the
minimax idea is presented in Section 3. In Section 4, the
simulation experiments are carried out to explain the
effectiveness of the proposed method. Finally, the conclusion is
given in Section 5.
2 TCP/AQM model
Consider the following dynamic model of TCP/AQM network system
which was presented in [5].
1 ( ) ( ( ))( ) ( ( ))
( ) 2 ( )
( )( ) ( ) ( ) ( ) > 0
( )
( )( ) p
W t W t R tW t p t R t
R t R t
N tq t W t C t q t
R t
q tR t T
C
(1)
where, max( ) [0, ]W t W is the window size of TCP source, max(
) (0, ]q t q is the instantaneous queue length of the
router, N(t) is the TCP network load, C(t) is the link
bandwidth, R(t) is the round-trip time, Tp is the propagation
delay, and p(t) is packet drop probability and takes value in
the interval [0,1]. Here, we suppose that N(t) and C(t)
are the constants, and it is worth noting that C(t) is an
unknown constant, which needs to be estimated later.
Therefore, N(t) and C(t) can be simply re-written as N and
C.
Remark 1. In this paper, the considered model is nonlinear
instead of linearized model, which can describe the
network more accurate. Therefore, the distorting factors in
Introduction stands for the inherent nonlinear, that is,
( ) ( ( )) 2 ( )W t W t R t R t and ( ) ( )/ ( )N t W t R t in
(1).
-
Regard the queue length and the window size as the state
variables and the congestion indication probability as
the control input of the network system, respectively. And
let
1 ( ) dx q t q , 2 ( )x W t , ( ) ( )u t p t .
where dq is the desired queue length.
Denote T1 2=( , )x xx . By neglecting system delay, the system
(1) can be rewritten as the following form.
1 2
2
1
( )
( , ) ( , ) ( )
x x
x a t x C
x f t g t u t
y kx
(2)
where, 0k is a design parameter, and
( )( )
Na t
R t ,
1( , )
( )f t
R tx ,
2
2( , )2 ( )
xg t
R t x ,
Remark 2. As we can see, equation (2) possesses a triangular
form. Based on the reference [17], the nonlinear
system with triangular structure is a kind of special nonlinear
systems, which is controllable, so stabilizable as long as it
does not contain zero-dynamics. As a matter of fact, the TCP/AQM
network model considered in the paper is a second
order system, which can be changed into a triangular system
without zero-dynamics. It is minimal for any time instants.
To the best of our knowledge, for backstepping design, there
have no reports about the effect of non-minimal behaviors
on the system performance and stability.
If we consider the situation with the UDP flow interference [1],
an interference item should be added to the
above system. Then, one has
1 ( ) ( ( ))( ) ( ( )) ( , , )
( ) 2 ( )
( )( ) ( )
( )
W t W t R tW t p t R t U t W q
R t R t
N tq t W t C
R t
(3)
where, U (t, W, q) is the UDP flow interference. Then, the state
space model of (3) can be changed into following
form:
1 2
2
1
( )
( , ) ( , ) ( ) ( , )
x a t x C
x f t g t u t t
y kx
x x x (4)
where term ( , )t x represents the system disturbance, i.e. the
UDP flow interference U (t, W, q).
For the case that there are both external disturbances (UDP
flows) and internal parameters C uncertain in this
system, a robust adaptive backstepping method controller based
on the parameter mapping mechanism is
combined with the minimax theory to design the nonlinear network
system.
Definition 1 [43]: A class of nonlinear systems is called
strict-feedback structure if for each subsystem, the
nonlinear function if are related to current state ix and
previous states 1ix and independent of 1ix , ..., nx where
1ix =[ 1x , 2x ,..., 1ix ]. Moreover, this kind of systems is
also named lower triangular systems.
-
Assumption 1: The values of the upper and lower bounds of the
uncertain parameter C are known, that is to
say min max( , )C C C , minC and maxC are the empirical values
from practice.
For (4), the interference suppression control problem can be
summarized as follows : For any given
circumstance, the impact of the interference generated by the
UDP flow and the uncertainty of the parameter C, on
the basis of full consideration of the value range of the
unknown parameters, the adaptive feedback controller
makes the following dissipation inequality
2 22
0( ( )) ( (0)) ( )
T
V x t V x y dt (5)
for all 0T be established, and the system is asymptotic stable,
at the same time the system 2L gain is less than
or equal to the interference suppression constant .
The objective of this work is to design a controller to make the
queue tend to its reference value, i.e., stabilizing
the TCP network system (2) or (4) at the desired queue.
3 Adaptive backstepping controller design
With the help of the model in Section 2, the corresponding
congestion controllers will be designed for the
systems (2) and (4) in this section.
3.1 Without interference and C is a known constant
According to the design idea of the backstepping method, define
the state transformation as follows
1 1
*
2 2 2
e x
e x x
(6)
Consider the subsystem with state variable1x of the system (2).
Then the subsystem becomes
*
1 1 2 2 2( ) ( )( )e x a t x C a t e x C (7)
where, *2x represents the virtual control variable of the
subsystem (7), 2e represents the error variable between the
system state2x and the virtual control
*
2x .
Step 1. The aim of this step is to design virtual feedback
control *2x making 1 0e . Thus, select the following
Lyapunov function
2
1 1
1.
2V e (8)
The derivative of V1 along the trajectory of the subsystem (7)
is given as follows
* *1 1 1 1 2 2 1 2 1 2 1[ ( )( ) ] ( ) ( ) .V e e e a t e x C a
t e e a t e x e C (9)
Choose an appropriate virtual control law as
-
*
2 1 1
1( )
( )x l e C
a t (10)
where l1 is a positive design parameter.
Substituting (10) into (9) yields
2
1 1 1 1 2( )V l e a t e e (11)
If 2 =0e , we can see that
2
1 1 1 0 V l e and the state 1e is asymptotic stable. In general,
the state 2e cannot be
guaranteed to be zero. So we need to consider the subsystem with
state variable x2 and design the control u to
make the error state variable 2e have the expected asymptotic
stability.
Step 2. The time derivative of 2e is
* 12 2 2 1 2( ) ( , ) ( )+
( )
l Ce x x f t g t u t l x
a t x (12)
Construct a Lyapunov function
2
2 1 2
1,
2V V e
. (13)
The derivative of V2 is
2 1 2 2
2
1 1 1 2 2 1 2
2
1 1 2 1 2 1 2
= ( ) [ ( , ) ( , ) ( ) ]( )
[ ( ) ( , )] [ ( , ) ( ) ].( )
V V e e
Cl e a t e e e f t g t u t l x
a t
Cl e e a t e f t e g t u t l x
a t
x x
x x
(14)
The control law is designed as
1 1 2 2 2
1( ) [ ( ) ( , ) ]
( , ) ( )
Cu t a t e f t l x l e
g t a t x
x (15)
where l2 is a positive design parameter. We can get
2 2
2 1 1 2 2 0V l e l e (16)
It follows from Lasalle Theorem and (16) that e1 and e2 converge
to zero. From the analytic induction above,
we know that the system (2) is asymptotic stable under the
designed controller, and then the system (1) realizes
the active queue management congestion control.
3.2 With UDP flow interference and C is an unknown constant
Consider the subsystem with state variable 1x of the system (4).
Define the state transformation is the same as (6).
The subsystem becomes
*
1 2 2 2( ) ( )( )e a t x C a t e x C (17)
-
where, *2x represents the virtual control variable of the
subsystem (17). The meaning of 2e is the same as one in
(6). The backstepping design process is divided into two
steps.
Step1. Select the following Lyapunov function
2
1 1
1.
2V e (18)
Differentiating V1 yields
* *1 1 1 1 2 2 1 2 1 2 1[ ( )( ) ] ( ) ( ) .V e e e a t e x C a
t e e a t e x e C (19)
Choose an appropriate virtual control law as
*
2 1 1
1 ˆ( )( )
x l e Ca t
(20)
where1 0l is a design parameter, Ĉ is the estimated value of
parameter C . Define the estimation error
ˆC C C . Then 2
1 1 1 1 2 1( ) .V l e a t e e e C (21)
*
2 2 2 1 2 1
1 ˆ( ) ( , ) ( ) ( , )+( ) ( )
Ce x x f t g t u t t l x l C
a t a t x x (22)
Step 2. The auxiliary variable C is introduced when constructing
the Lyapunov function to ensure that the
process of parameter estimation always occurs in min max( , )C C
. Therefore, the constructive augmented Lyapunov
function is given as follows :
2 2 2
2 1 2
1 1 ˆ( ) ( )2 2
V V e C C C C
(23)
where 0 is a design parameter.
The derivative of V2 is computed as follows.
2
2 1 1 1 2 1 2 2
2 11 1 1 2 1 1 2
1 ˆ ˆ= ( ) ( )
1 1ˆ ˆ ˆ= + [ ( ) ] ( )( ) ( )
V l e a t e e e C e e CC C C C
ll e e C e a t e f gu l x C C CC C C C
a t a t
(24)
The unknown interference term ( , )t x appears in equation (24).
So the control law cannot be directly designed
like as equation (15). The disturbance needs to be addressed
before the controller is designed. Generally, two
ways can be used to deal with disturbances. One is that the
upper-bound of interference is assumed artificially.
The interference, however, is often difficult to measure
accurately and is often uncertain. So it is difficult to get a
proper upper-bound in practice. Therefore, this way may not be
in accordance with physical reality [13]. The other
way is to reduce the interference related items in the Lyapunov
function during the design of controller, which
will lead to hypothetical condition enhancement [44].
To avoid the limitations and conservatism mentioned above, prior
to the design of the controller, we first deal
with the interference terms based on the minimax theory. That
is, the worst effect of interference on the system
will be calculated.
Consider the following index performance
2 22
0
1( )d
2J y t
(25)
-
where, 0 is the interference suppression constant. In order to
calculate the worst interference, the following
test function is constructed
2 222
1( )
2V y (26)
Remark 3. With the aid of the existing results [45]-[46], the
test function (26) is designed by combining the
Lyapunov function and index performance (25), and its
construction method is not sole, the case of which is
similar to the choice of Lyapunov function.
Substituting (24) into (26) yields
2 2 2 211 1 1 2 1 1 2
2 2 2 211 1 1 2 1 1 2
1 1 1ˆ ˆ ˆ[ ( ) ] ( ) ( )( ) ( ) 2
1 1 1 1ˆ ˆ ˆ( ) [ ( ) ] ( ) .2 ( ) ( ) 2
ll e e C e a t e f gu l x C C CC C C C y
a t a t
ll k e e C e a t e f gu l x C C CC C C C
a t a t
(27)
Here, 21
10
2l k . The first derivative of about is
2
2 .e
(28)
Let it be zero. We obtain
*
22
1( , ) .t e
x (29)
Then, we can get the second derivative of , which is less than
zero.
22
20.
(30)
It can be seen, from the discussion above, that has the maximum
value at *( , )t x , i.e.
* 2 2 * 2
2
1max ( ) ( ( ) ).
2V y
(31)
For the test function (26), the maximum value is
2 22
2
1max max[ ( )].
2V y
(32)
Integrating both sides of (32) simultaneously yields
2 22
20 0 0
1max d max[ d ( )d ]
2t V t y t
(33)
Denote
0dt
(34)
Then
2 2max max[ ( ) (0) ]V V J
(35)
It means that max
is equivalent to max J
. Therefore, *( , )t x makes the critical function get the
maximum
and then the index performance J get the maximum. So we can say
that *( , )t x is the worst disturbance for the
system.
-
Substituting the worst interference *( , )t x into the equation
(27), we have
2 2 11 1 1 2 1 2 1 22
1 1 1 1ˆ ˆ ˆ( ) ( ) .2 2 ( ) ( )
ll k e e C e a t e f gu e l x C C CC C C C
a t a t
(36)
The control law is designed as
12 2 1 1 22
1 1 1ˆ ˆ( ) ( )2 ( ) ( )
lu t l e a t e f l x C C
g a t a t
(37)
where l2 is a positive design parameter. Substituting u to
yields
2 2 2 11 1 2 2 1 2
1 1 ˆ ˆ( ) .2 ( )
ll k e l e e e C CC C C C
a t
(38)
Select an adaptive law
11 2
ˆ( ).( )
lC e e C C
a t
(39)
where 0 is a design parameter. Through the mapping of C , the
estimated value Ĉ of the unknown parameter
C can be obtained by
min max
min min
max max
( , ),
ˆ ,
.
C if C C C
C C if C C
C if C C
(40)
By choosing a suitable value, whenmaxC C or minC C , it can pull
C back to a predefined range min max( , )C C .
So, we can get
2 2 2
1 1 2 2
10
2l k e l e
(41)
Define Lyapunov function ( )V x as 22 ( )V x . If 0 , then ( )
0V x ; if 0 , then the 2L gain from the
disturbance input to the controlled output is less than and
equal to . It follows from Lasalle Theorem that 1e and
2e are globally asympototically stable.
From the above presented induction analytic synthesis, we know
that the system (4) is globally asymptotically
stable under the designed controller, and then the system (3)
with the UDP flow interference has realized the
active queue management congestion control.
Theorem 1. For the given disturbance attenuation constant 0 , if
there exist the design parameters 0k ,
2
1
1
2l k and 2 0l , the L2 disturbance attenuation problem of system
(4) can be solved by feedback control law
(37) and parameter update laws (39) and (40), then the system is
globally asymptotically stable, and a positive
storage function ( )V x exists such that the dissipation
inequality (5) holds for any final time T, and the closed-loop
system is characterized with disturbance rejection.
In the process of designing the adaptive law, by introducing the
parameter mapping mechanism, the parameter
estimation range is limited to the specified range, which avoids
the process of parameter search occurring outside
the true value range, and greatly increases the estimation
efficiency.
-
Remark 4. It should be pointed out that Zeno behavior may occur
in the process of controller design and it will
affect the simulation results and applicaions to some extent.
Hence, this problem deserves to be addressed in
future works due to its importance and significance.
Remark 5. Due to the use of the backstepping technique, the
computational complexity becomes a main the
limitation of this method. It is well known that the dynamic
surface control (DSC) can be employed to solve the
problem of "explosion of complexity" of backstepping. As a
result, the proposed scheme can be further improved
by DSC. Besides, it is worth noting that the model of TCP/AQM
network is a second-order system, hence, the
corresponding computation is less complex.
4 Simulation experiments
According to the design process of the controllers, some
simulation results are given to verify the effectiveness
of the presented method. The system parameters and design
parameters are selected as:
d60, 1750packets s, 100packets, N C q (42)
1 2=0.8, 1, 1.5, 1, 1.2, 1k l l , min max1600packets s,
2000packets s C C .
Remark 6. It is worth noting that the parameters in (42) are
decided by the specific network, that is to say that
different networks have different values. In this work, the
selected parameter values are the same with that in [23].
Without the external disturbance, the adaptive laws with
different proportion delays are shown in Figure 1. It
can be observed from Figure 1 that the adaptive law tracks the
desired value 1750 packets/s in a short period of
time in different propagation delays. Specifically, the Ĉ
reaches to 1750packets/s within 2s when Tp=0.2s,
however, it converges to the same value within 0.5s when
Tp=0.1s. Besides, it follows from the above analysis
that the faster Ĉ converges, the smaller the Tp is.
In Figure 2 and Figure 3, simulation results considering UDP
flow interference in the system are shown. The
control law u is shown in Figure 2, in which the left figure is
the control law u in round-trip delays R(t)=250ms,
the packet loss rate is between 0 and 0.005 during a very short
time, and the right figure is the control law u in
round-trip delays R(t)=300ms, the packet loss rate is between
0.005 and 0.01 in a very short period of time, that is,
the probability of packet loss is very small. The adaptive law
Ĉ is shown in Figure 3, in which the left figure is the
adaptive law in Tp=0.2s, a four-second rising oscillation
returns to the true value, and the right figure is the
adaptive law in Tp=0.1s, a two-second rising oscillation returns
to the true value. The adaptive controller designed
with the parameter mapping mechanism can quickly limit the
parameter estimation value to the specified range.
The simulation results explain the effectiveness of the designed
adaptive interference suppression backstepping
controller.
Remark 7. In adaptive controller design, the tuning parameters
are and . It follows from [47] that should
be increased and should be decreased to reduce the radius of
neighborhood and accelerate the convergence rate
of the variables. However, if is large, the control energy is
large. Therefore, in practical applications, the design
parameters should be adjusted carefully for achieving suitable
transient performance and control action.
Besides, in order to explain the superiority of the presented
approach, the simulation comparison is made
between the proposed method and the adaptive backstepping H
techniques by considering two different
propagation delays. The parameters of the proposed controller
are given as follows: 1 2=0.5, 1, 2.5, 1,k l l
-
0.2, 0.5 . The parameters of the adaptive backstepping H are
chosen as the same with [23], that is,
1 25,c 2 15,c 1 24, 1 , 10, 1, 100packets C refq , 0 0.2, 0.01,
1 . Other parameters are
the same with (42). In Figure 4, under the same interference
condition, the left figure is the propagation delays in
Tp=0.1s. The controller proposed in this paper makes the queue
tracking error1e to converge to zero after about
1.5s, and the controller of adaptive backstepping H method makes
the queue tracking error 1e converge to zero in
6s. It can be seen from the right figure of Figure 4 that the
proposed controller makes the queue tracking error
1e converge to zero about 2.5s. However, under the large delay
of adaptive backstepping H techniques, the queue
tracking error1e always fluctuates above and below the zero
point. In order to illustrate the simulation results
better, Table I is given, in which the following four cases are
discussed. Case 1. The response time of Ĉ without
the disturbance; Case 2. The response time of Ĉ with the
disturbance; Case 3. The convergent time of the queue
tracking error by employing the proposed scheme; Case 4. The
convergent time of the queue tracking error by
using H control method.
Table I
Case 1 Case 2 Case 3 Case 4
Tp=0.1s t = 0 . 5 s t = 2 s t =1.5s t =5s
Tp=0.2s t = 2 s t = 4 s t =2.5s t =7s
Figure 1: Left is the adaptive law in Tp=0.2s and right is the
adaptive law in Tp=0.1s.
-
Figure2: Left is the control law u in round-trip delays
R(t)=250ms and right is the control law u in round-trip delays
R(t)=300ms.
Figure 3: Left is the adaptive law in Tp=0.2s and right is the
adaptive law in Tp=0.1s of the system with disturbance.
Figure 4: Left is the e1 in Tp=0.1s and right is the e1 in
Tp=0.3s of the system with disturbance
5 Conclusions
In this work, the congestion control problem is solved for a
class of nonlinear TCP network systems by using
adaptive backstepping technology and the minimax method. An AQM
control algorithm is presented to avoid the
influence of retracting interference terms and neglecting
feature of systems, in which the adaptive controller is
designed based on the parameter mapping mechanism. To the worst
interference calculated with the minimax
method, a robust controller is designed. The simulation results
show that the proposed method has strong
robustness, and can get the faster system response and smaller
overshoot. The proposed method can be combined
with the fuzzy logic control (see [48] and references therein)
to study the related congestion control problem in
future research activities. In addition, motivated by [49], the
congestion control can be considered for discrete-
time network systems.
Data Availability
The data used to support the findings of this study are
available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest
regarding the publication of this article.
Acknowledgement
-
This work is supported by the National Natural Science
Foundation of China [grant number 61773108], Liaoning
Natural Science Fund Project [grant number 20170540788],
Educational Commission of Liaoning Province [grant
number L2015198] and Science Foundation for Doctorate Research
of Liaoning Province [grant number 201601091].
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Highlight
The UDP flow is addressed by the minimax method instead of H
control; A novel congestion
algorithm is presented to solve an adaptive tracking problem;
The steady-state error of the
closed-loop system satisfies the design requirements.
*Highlights (for review)
-
Conflicts of Interest
The author declares that there are no conflicts of interest
regarding the publication of this article.
*Conflict of Interest
Design of adaptive backstepping congestion controller for TCP
networks with UDP flows based on minimax
