# Adaptive Finite-Time Congestion Control for Uncertain TCP/AQM Network with Unknown Hysteresis

Authors: Weimin Zheng; Yanxin Li; Xiaowen Jing; Shangkun Liu
Journal: Complexity
 (2020)

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## Abstract

The issue of adaptive practical finite-time (FT) congestion control for the transmission control protocol/active queue management (TCP/AQM) network with unknown hysteresis and external disturbance is considered in this paper. A finite-time congestion controller is designed by the backstepping technique and the adaptive neural control method. This controller guarantees that the queue length tracks the desired queue in finite-time, and it is semiglobally practical finite-time stable (SGPFS) for all the signals of the closed-loop system. At last, the simulation results show that the control strategy is effective.


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## Body

## 1. Introduction

Recently, the communication network based on TCP has been rapidly developed and widely used. However, the congestion of network traffic has become a critical problem in network control. TCP congestion control can not only ward off network collapse and avert locking behavior, but also effectively reduce the probability of control loop synchronization [1]. It is of great significance to maintain the stability and robustness of the TCP network. Since Jacobson proposed the end-to-end TCP congestion control algorithm in 1988 [2], many scholars have conducted more in-depth and detailed research, such as Vegas [3], Sack [4], and New Reno [5], and they all concentrate on end-to-end congestion control based on the end system. However, with the increasing demand of application and the improvement of service quality, the TCP congestion control mechanism of end-to-end is no longer suitable. After that, some scholars have proposed a solution called AQM [6], which is one of the most widely studied congestion solutions at present. It can drop or mark some packets before the router generates a full queue state, so that the TCP source can timely sense the network congestion state and take corresponding measures. The first algorithm is named as random early detection (RED) [7], but RED and its improved algorithms [8, 9] are too complicated for parameter configuration. Based on fluid flow theory, Misra et al. proposed a nonlinear model of TCP/AQM in 2000 [10]. Based on the above model, many researchers have combined control theory to design several congestion schemes, such as P and PI [11], PD [12], and PID [13]. An AQM algorithm is presented on the strength of the fuzzy sliding-mode control method in the nonlinear control method in [14], which improves the control effect of the system. In [15], considering the limited input of the TCP network system, an AQM algorithm is presented on the strength of the sliding-mode control method, which makes the system obtain better asymptotic stability. Tan et al. [16] proposed a congestion control scheme composed of source and link algorithms. Zhang et al. [17] analyzed the TCP/RED model from a time-delay control theory standpoint, and the time-delayed control analyzing techniques explored can be extended to other AQM or AQM-based schemes for their stability analysis. The primal-dual algorithm has been analyzed from the multivariable time-delay control theory standpoint in [18]. The stability analysis can be applied to various TCP/AQM systems other than FAST TCP/DropTail and TCP/AVQ. The AQM controller with specified performance is designed by the backstepping method considering the interference of UDP flow [19]. Due to the complex environment and continuous application of the network [20–22], congestion algorithms need to further improve network performance and achieve better congestion control effects.Nowadays, many scholars have noticed the hysteresis phenomenon widely existing in the nonlinear system. In the process of practical application, the tracking performance of the system has been limited by the hysteresis, which even makes the system unstable [23, 24]. To reduce the impact of nonlinear system control on the unknown hysteresis in the actuator, more and more scholars have studied the design of the controller in [25–28]. Zhou et al. [25] not only first proposed a new Bouc–Wen hysteresis model, but also put forward an adaptive control method to ensure a good tracking performance by constructing an inverse of compensating hysteresis nonlinearity. For the sake of the stability of the controlled system, Su et al. [26] designed an adaptive control scheme based on the backlash-like hysteresis model. Wang et al. [28] discussed the adaptive stabilization of pure-feedback nonlinear systems, and it not only solved the unknown direction hysteresis, but also eliminated the constraint assumption that the nonlinear function needs to satisfy the linear growth condition by using the characteristic of the Nussbaum function and introducing a virtual controller. Up to now, there are no literature studies for the congestion control of TCP/AQM networks with unknown hysteresis.On the other hand, some scholars pay more attention to the FT control because the FT stability has more meaning than infinite-time stability in practical application [29–31]. In practice, the control goal is promising to be realized in a limited time, and the control scheme of infinite time cannot achieve such a goal because they will lead to a long time transient response. The FT stability is different from the asymptotic stability of infinite time, its control method will enable the system to achieve the transient performance quickly, and in FT, the system state variables can be converged in equilibrium. Dorato conducted a comprehensive study on the problem of FT stability and elaborated on the differences between FT stability and asymptotic stability [29]. The Lyapunov theory of FT stability was first put forward in [30, 32]. The authors in [33, 34] proposed some FT control schemes for the nonlinear systems via the Lyapunov stability theory. However, in practice, these FT control strategies cannot satisfy the actual control system with unknown nonlinearity. Wang et al. [34] studied the FT tracking problem with unknown functions. A criterion of SGPFS is set up for the first time by the fuzzy logic system (FLS), and a novel adaptive fuzzy control strategy based on it is presented.To sum up, this manuscript researches FT congestion control for the TCP/AQM network with unknown hysteresis and external disturbance. The main works are summarized as follows:(1)
Inspired from [35], a newfangled model of TCP is established, which considers the effects of hysteresis input and exogenous disturbance. The model in this work is more general and more exact.(2)
An adaptive FT congestion controller is constructed by combining the backstepping technique and the radial basis function neural networks (RBFNNs) in this paper. This controller guarantees that the queue length tracks the desired queue in FT, and all the signals of the closed-loop system are SGPFS.(3)
The classical FT control strategy requires that the nonlinear function of the controlled system must satisfy the linear growth condition or matching condition [30, 36–38]. In this article, as the nonlinear function is unknown, the studied system cannot satisfy the linear growth condition. Therefore, in the new congestion control strategy, continuity is the only requirement for nonlinear functions. Hence, this scheme is more general.The surplus of the manuscript is summarized as follows. A network model and advance preliminaries are recommended in Section2. The main result is shown in Section 3. Section 4 gives a simulation example. Finally, Section 5 draws a conclusion.


## 2. Model and Preliminaries

### 2.1. TCP/AQM System Model

This article, based on [35], considers the following TCP/AQM network, in which the authors takes account of external interference, and in contrast, the time-delay is neglected:(1)W˙t=1Rt1−pt−Wt2Wt−RtRt−Rtpt−Rt,q˙t=−Ct+NtRtWt+ωt,q>0max0,−Ct+NtRtWt+ωt,q=0,Rt=qtCt+Tp,where Wt∈0,Wmax is the TCP window size, qt∈0,qmax is the queue length of the router, Rt is the round-trip time, Ct, Nt, and TP are the available link capacity, the number of TCP sessions, and the propagation delay, respectively, p∈0,1 is the probability of packet loss, and ωt is the external disturbance, which can be thought of as unresponsive flows.Assumption 1.
In fact, most Internet routing scenarios change over time slowly. Therefore, we suppose that the parametersNt and Ct are fixed values [13, 39]. Nt, Ct, and Rt can be simply rewritten as N, C, and R.Assumption 2.
It is bounded for the external disturbanceωt and its derivative.
According to [40], the rate model is adopted by(2)r˙t=w˙t−rtR˙R,with rt=Wt/R. It is not hard to conclude that, based on (1) and (2),(3)r˙t=1R2−1R2pt−r2t2pt+rtR−Nr2tRC−rtωtRC,q˙t=−C+Nrt+ωt.
Setx1=qt, x2=rt, and ut=pt, and the above model can be converted to(4)x˙1=Nx2+ωt−C,x˙2=fx+gxut+ψxωt,y=x1,where xt=x1t,x1tT and fx and gx are as follows:(5)fx=1R2−NRCx2+x2R,gx=−1R2−12x22,ψx=−x2RC.


### 2.2. Hysteresis Nonlinearity

Consider a modified Bouc–Wen hysteresis in the form of(6)Hu=μ1u+μ2ι.It is assumed that signμ1 = sign μ2 and(7)ι˙=u˙−ϖu˙ιϑ−1−ξu˙ιϑ=u˙fι,u˙,ιt0=0,where ϖ>ξ, n>1, ι≤ι¯=1/ϖ+ξn, ϖ is the shape and amplitude of the hysteresis, and ϑ represents the smoothness from the initial slope to the asymptote’s slope. Define fι,u˙ as(8)fι,u˙=1−signu˙ϑιϑ−1ι−ξιϑ.Remark 1.
The sign ofμ1 determines the direction of hysteresis.Assumption 3.
Without losing generality, assume thatsignμ1>0.Remark 2.
In the process of practical application, the tracking performance of the system has been limited by the hysteresis, which even makes the system unstable. For the TCP/AQM system model (4), the authors take account of the hysteresis nonlinearity in the control system to reduce the influence of the hysteresis input on the system and realize the stability of the controlled system.


### 2.3. RBFNNs

RBFNNs are applied to identify the unknown nonlinear functions. The RBFNNs can be written as(9)fnnς=ϒTℏς,where ς∈Ως⊂ℛq is the input vector, ϒ=ϒ1,ϒ2,…,ϒℵT∈ℛℵ is the weight vector, ℵ>1 represents the number of RBFNNs nodes, and ℏς=ℏ1ς,ℏ2ς,…,ℏℵςT∈ℛℵ denotes the basis function vector. Select the Gaussian basis function ℏiς as follows:(10)ℏiς=exp−ς−κiTς−κiν2,i=1,2,…,ℵ,where ν is the width of the Gaussian function and κi=κi1,…,κiq is the center of the receptive field.Lemma 1 (see [41]).
Define a continuous functionfς on a compact set Ω. Then, for ∀ε>0, the following holds:(11)fς=ϒ∗Tℏς+ϵς,where(12)Φ=argminϒsupς∈Ωfx¯−ϒTℏς,ϵς≤ε.Definition 1.
(see [42]). Consider the following system:(13)℘˙=f℘,where ℘ represents the state vector. The origin of system (13) is SGPFS if all ℘t0=℘0, and there exist δ>0 and a setting time 0<T℘0,δ<∞ such that ℘t≤δ when t≥t0+T.Assumption 4.
There exists an unknown constantb2>0 such that(14)b2≤gx.Lemma 2 (see [43]).
For any real variablesx0andy0 and constants ı>0,ξ>0,andμ>0, the following inequality is satisfied:(15)ϕξψξ≤ıı+ξμϕı+ξ+ξı+ξμ1/μψı+ξ.Lemma 3 (see [41]).
For any real numberli,i=1,2,…,n, when ζ∈0,1 and λ∈0,2, it holds(16)l1+⋯lnζ≤l1ζ+⋯+lnζ,l12+⋯+ln2λ≤l1λ+⋯+lnλ2.Lemma 4 (see [44]).
Consider system (13), and suppose that there is a C1 function V℘ and V℘ with V0=0 is positive definite on D, one has(17)V˙℘≤−τVζ℘+ϱ,where τ>0,0<ζ<1, and 0<ϱ<∞. Then, the trajectory of system ℘˙=f℘ is SGPFS.Proof.
According to equation (17), for 0<∅<1, the following inequality holds:(18)V˙℘≤−∅τVζ℘−1−∅τVζ℘+ϱ.
LetΓ℘=℘Vζ℘≤ϱ/1−∅τ and Γ˜℘=℘Vζ℘>ϱ/1−∅τ. Discuss the following two cases:(1)
If the initial value satisfies℘t∈Γ˜℘, then one has(19)V˙℘≤−∅τVζ℘.According to equation (19), it holds that(20)∫t0TV℘.Vζdt≤∫t0T∅τdt.Then, we can get the following inequality:(21)11−ζV1−ζ℘T−11−ζV1−ζ℘t0≤−∅τT.Let(22)Tr=11−ζ∅τV1−ζ℘t0−ϱ1−∅τ1−ζ/ζ.Then, for∀t≥T, combining equations (21) and (22) can be expressed as(23)℘t∈Γ℘.(2)
If the initial value satisfies℘t∈Γ℘, then according to the first case, the trajectory of ℘t does not exceed Γ℘. Then, it can be obtained that there exists Tr<∞, ∀t≥T, such that ℘t∈Γ℘. That is, the solution of the nonlinear system ℘˙=f℘ is SGPFS.


## 2.1. TCP/AQM System Model

This article, based on [35], considers the following TCP/AQM network, in which the authors takes account of external interference, and in contrast, the time-delay is neglected:(1)W˙t=1Rt1−pt−Wt2Wt−RtRt−Rtpt−Rt,q˙t=−Ct+NtRtWt+ωt,q>0max0,−Ct+NtRtWt+ωt,q=0,Rt=qtCt+Tp,where Wt∈0,Wmax is the TCP window size, qt∈0,qmax is the queue length of the router, Rt is the round-trip time, Ct, Nt, and TP are the available link capacity, the number of TCP sessions, and the propagation delay, respectively, p∈0,1 is the probability of packet loss, and ωt is the external disturbance, which can be thought of as unresponsive flows.Assumption 1.
In fact, most Internet routing scenarios change over time slowly. Therefore, we suppose that the parametersNt and Ct are fixed values [13, 39]. Nt, Ct, and Rt can be simply rewritten as N, C, and R.Assumption 2.
It is bounded for the external disturbanceωt and its derivative.
According to [40], the rate model is adopted by(2)r˙t=w˙t−rtR˙R,with rt=Wt/R. It is not hard to conclude that, based on (1) and (2),(3)r˙t=1R2−1R2pt−r2t2pt+rtR−Nr2tRC−rtωtRC,q˙t=−C+Nrt+ωt.
Setx1=qt, x2=rt, and ut=pt, and the above model can be converted to(4)x˙1=Nx2+ωt−C,x˙2=fx+gxut+ψxωt,y=x1,where xt=x1t,x1tT and fx and gx are as follows:(5)fx=1R2−NRCx2+x2R,gx=−1R2−12x22,ψx=−x2RC.


## 2.2. Hysteresis Nonlinearity

Consider a modified Bouc–Wen hysteresis in the form of(6)Hu=μ1u+μ2ι.It is assumed that signμ1 = sign μ2 and(7)ι˙=u˙−ϖu˙ιϑ−1−ξu˙ιϑ=u˙fι,u˙,ιt0=0,where ϖ>ξ, n>1, ι≤ι¯=1/ϖ+ξn, ϖ is the shape and amplitude of the hysteresis, and ϑ represents the smoothness from the initial slope to the asymptote’s slope. Define fι,u˙ as(8)fι,u˙=1−signu˙ϑιϑ−1ι−ξιϑ.Remark 1.
The sign ofμ1 determines the direction of hysteresis.Assumption 3.
Without losing generality, assume thatsignμ1>0.Remark 2.
In the process of practical application, the tracking performance of the system has been limited by the hysteresis, which even makes the system unstable. For the TCP/AQM system model (4), the authors take account of the hysteresis nonlinearity in the control system to reduce the influence of the hysteresis input on the system and realize the stability of the controlled system.


## 2.3. RBFNNs

RBFNNs are applied to identify the unknown nonlinear functions. The RBFNNs can be written as(9)fnnς=ϒTℏς,where ς∈Ως⊂ℛq is the input vector, ϒ=ϒ1,ϒ2,…,ϒℵT∈ℛℵ is the weight vector, ℵ>1 represents the number of RBFNNs nodes, and ℏς=ℏ1ς,ℏ2ς,…,ℏℵςT∈ℛℵ denotes the basis function vector. Select the Gaussian basis function ℏiς as follows:(10)ℏiς=exp−ς−κiTς−κiν2,i=1,2,…,ℵ,where ν is the width of the Gaussian function and κi=κi1,…,κiq is the center of the receptive field.Lemma 1 (see [41]).
Define a continuous functionfς on a compact set Ω. Then, for ∀ε>0, the following holds:(11)fς=ϒ∗Tℏς+ϵς,where(12)Φ=argminϒsupς∈Ωfx¯−ϒTℏς,ϵς≤ε.Definition 1.
(see [42]). Consider the following system:(13)℘˙=f℘,where ℘ represents the state vector. The origin of system (13) is SGPFS if all ℘t0=℘0, and there exist δ>0 and a setting time 0<T℘0,δ<∞ such that ℘t≤δ when t≥t0+T.Assumption 4.
There exists an unknown constantb2>0 such that(14)b2≤gx.Lemma 2 (see [43]).
For any real variablesx0andy0 and constants ı>0,ξ>0,andμ>0, the following inequality is satisfied:(15)ϕξψξ≤ıı+ξμϕı+ξ+ξı+ξμ1/μψı+ξ.Lemma 3 (see [41]).
For any real numberli,i=1,2,…,n, when ζ∈0,1 and λ∈0,2, it holds(16)l1+⋯lnζ≤l1ζ+⋯+lnζ,l12+⋯+ln2λ≤l1λ+⋯+lnλ2.Lemma 4 (see [44]).
Consider system (13), and suppose that there is a C1 function V℘ and V℘ with V0=0 is positive definite on D, one has(17)V˙℘≤−τVζ℘+ϱ,where τ>0,0<ζ<1, and 0<ϱ<∞. Then, the trajectory of system ℘˙=f℘ is SGPFS.Proof.
According to equation (17), for 0<∅<1, the following inequality holds:(18)V˙℘≤−∅τVζ℘−1−∅τVζ℘+ϱ.
LetΓ℘=℘Vζ℘≤ϱ/1−∅τ and Γ˜℘=℘Vζ℘>ϱ/1−∅τ. Discuss the following two cases:(1)
If the initial value satisfies℘t∈Γ˜℘, then one has(19)V˙℘≤−∅τVζ℘.According to equation (19), it holds that(20)∫t0TV℘.Vζdt≤∫t0T∅τdt.Then, we can get the following inequality:(21)11−ζV1−ζ℘T−11−ζV1−ζ℘t0≤−∅τT.Let(22)Tr=11−ζ∅τV1−ζ℘t0−ϱ1−∅τ1−ζ/ζ.Then, for∀t≥T, combining equations (21) and (22) can be expressed as(23)℘t∈Γ℘.(2)
If the initial value satisfies℘t∈Γ℘, then according to the first case, the trajectory of ℘t does not exceed Γ℘. Then, it can be obtained that there exists Tr<∞, ∀t≥T, such that ℘t∈Γ℘. That is, the solution of the nonlinear system ℘˙=f℘ is SGPFS.


## 3. Main Results

### 3.1. Adaptive RBFNN Controller Design

Before we begin designing, coordinate transformation is introduced as follows:(24)z1=x1−qd,z2=x2−α1,where α1 is a virtual controller defined as(25)αi=b1N−12ai2ziθ^iℏiZiTℏiZi−12zi−kizi2ζ−1,where ζ>0, ki>0, and ai>0 are design parameters and the ith RBFNN ℏiZi can identify the unknown nonlinear function in the design process. Define the unknown constant θi=ϒi∗2/bi, where θ^i is the estimation of θi and the estimation error is θ˜i=θi−θ^i.Choose the control lawu, the auxiliary controller u¯, and the adaptive law as follows:(26)u=e^u¯,(27)u¯=−12ai2ziθ^iℏiZiTℏiZi−12zi−kizi2ζ−1,(28)θ^.i=ri2ai2zi2ℏiZiTℏiZi−βiθ^i,where ri>0 and σi>0. We assume that θ^it≥0 in this article, e^=1/μ^1 is the estimation of e^=1/μ^1, and e˜=e−e^ is the estimated error. Select e^˙ as(29)e^˙=−ϱznbnu¯−ϱ0e^,where ϱ and ϱ0 are the design constants.Step 1: Lyapunov function candidate will be chosen as follows:(30)V1=12z12+b12r1θ˜12.DifferentiatingV1 yields(31)V1.=z1Nx2+ωt−C−qd−b1r1θ˜1θ^.1.Choose the functionf^1Z1 defined by(32)f^1Z1=−C+ωt−y˙d+k1z12ζ−1,then equation (29) can be rewritten as(33)V˙1=z1f^1+Nx2−b1r1θ˜1θ^.1−k1z12ζ.According to Lemma1, RBFNN ϒ1∗Tℏ1Z1 can be applied to identify the unknown function f^1Z1. For any given ε1>0,(34)f^1Z1=ϒ1∗Tℏ1Z1+ϵ1Z1,ϵ1Z1≤ε1,whereϵ1Z1 is the approximation error. Consequently, let θ1=ϒ1∗2/b1, and one can obtain(35)z1f^1Z1=z1ϒ1∗TΥ1∗Υ1∗ℏ1−z1δ1≤b1z122a12Υ1∗2b1ℏ1Tℏ1+12a12+12b1z12+12b1ε12=b12a12z12θ1ℏ1Tℏ1+12a12+12b1z12+12b1ε12.Combining equations (31) and (33) gives(36)V˙1≤z1b12a12z1θ1ℏ1Tℏ1+Nα1+Nz1z2−b1r1θ˜θ^.1−k1z12ζ+12a12+12b1z12+12b1ε12.According to equation (24) and Assumption 4, it holds that(37)z1Nα1≤​−b12a12z12θ^1ℏ1Tℏ1​−k1b1z12ζ​−12b1z12.Combining (36) with (37) gives(38)V˙1≤b12a12z12θ1ℏ1Tℏ1−b12a12z12θ^1ℏ1Tℏ1−k11+b1z12ζ+Nz1z2−b1r1θ˜θ^.1+12a12+12b1ε12.Choose the adaptation law as(39)θ^.1=r12a12z12ℏ1Z1Tℏ1Z1−βiθ^1.According to equation (37), it holds that(40)b1r1θ˜θ^.1=b12a12z12θ1˜ℏ1Tℏ1−b1β1r1θ˜1θ^1.It is noted that(41)b1β1r1θ˜1θ^1≤−b1β12r1θ˜12+b1β12r1θ12.Therefore, combining equations (38), (40), and (41) can be expressed as(42)V˙1≤ℓ1+Nz1z2,where(43)ℓ1=−k11+b1z12ζ+b1β12r1−θ˜12+θ12+12a12+12b1ε12.Step 2: choose a Lyapunov function candidate as(44)V2=V1+12z22+b22r2θ˜22+μ12ϱe˜2.It is that(45)V˙2=ℓ1+Nz1z2−b2r2θ˜2θ^.2−μ1ϱe˜e^˙+z2fx+gxHu¯+ψxωt−α˙1.Establish the actual controller as(46)u¯=−12a22z2θ^2ℏ2Z2Tℏ2Z2−12z2−k2z22ζ−1.According to equations and Assumption4, one has(47)z2gxHu¯−μϱe˜e^˙≤z2b2u¯−z2b2μ1e˜u¯+z2b2μ2ι−μ1ϱe˜−ϱz2b2u¯−ϱ0e^≤z2b2u¯+z2b2μ2ι+μ1ϱe˜e^≤−12a22z22b2θ^2ℏ2Tℏ2−12b2z22−k2b2z22ζ+12b2z22+12μ2ι¯2+ϱ02ϱμ1,where ι¯=n1/ϑ+ξ.Now, substituting inequalities (47) into (45) results in(48)V˙2≤ℓ1+z2f^2−12a22z22b2θ^2ℏ2Tℏ2−12b2z22−k2b2z22ζ−k2z22ζ+12μ2ι¯2+ϱ02ϱμ1−b2r2θ˜2θ^.2,where f^2=fx+k2z22ζ+Nz1+ψxωt+1/2b22z22−α˙1, which can be estimated by RBFNN ϒ2∗Tℏ2Z2 for any given positive constant ε2. Similarly, let θ2=ϒ2∗2/b2, and one easily obtains(49)z2f^2Z2=z2ϒ2∗TΥ2∗Υ2∗ℏ2−z1δ2≤b2z222a22Υ2∗2b1ℏ2Tℏ2+12a22+12b2z22+12b2ε22=b22a22z22θ2ℏ2Tℏ2+12a22+12b1z22+12b2ε22.Then, equation (46) can be rewritten as(50)V˙2≤ℓ1+b22a22z22θ2ℏ2Tℏ2−12a22z22b2θ^2ℏ2Tℏ2−k21+b2z22ζ+12a22+12b2ε22+12μ2ι¯2+ϱ02ϱμ1−b2r2θ˜2θ^.2.


### 3.2. Stability Analysis

Theorem 1.
Consider system (4). Suppose Assumptions 2–4 hold, then all the signals are SGPFS, under any bounded initial conditions when controller (25) and adaptive law (28) are employed.Proof.
Choose the adaptation law as(51)θ^.2=r22a22z22ℏ1Z2Tℏ1Z2−βiθ^2.
According to equation (49), it holds that(52)b2r2θ˜θ^.2=b22a22z22θ2˜ℏ2Tℏ2−b2β2r2θ˜2θ^2.
It is noted that(53)b2β2r2θ˜2θ^2≤−b2β22r2θ˜22+b2β22r2θ22.
Substituting (52) and (53) into (50), it gives(54)V˙2≤ℓ1+b2β2r2θ˜2θ^2−k2b2z22ζ−k2z22ζ+12μ2ι¯2+ϱ02ϱμ1+12a22+12b2ε22.
Then, we can get(55)V˙2≤−∑i=12ki1+bizi2ζ+b1β12r1−θ˜12+θ12+b2β22r2−θ˜22+θ22+12a12+1b1ε12+12a22+1b2ε22+12μ2ι¯2+ϱ02ϱμ1.
Letkmin=min1≤i≤2ki1+bi, and by Lemma 3, it is deduced that(56)−∑i=12ki1+bizi2ζ≤−kmin∑i=12zi2ζ≤−K¯12∑i=12zi2ζ,where K¯=2ζkmin.
Letβmin=min1≤i≤2βi, and by taking equation (54) into account, inequality (55) can be rewritten as(57)V˙2≤−K¯12∑i=12zi2ζ−βminζ∑i=12bi2riθ˜i2ζ+∑i=12biβi2riθ˜i2ζ+b1β12r1−θ˜12+θ12+b2β22r2−θ˜22+θ22+12a12+1b1ε12+12a22+1b2ε22+12μ2ι¯2+ϱ02ϱμ1≤−λVζ+∑i=12biβi2riθ˜i2ζ+b1β12r1−θ˜12+θ12+b2β22r2−θ˜12+θ12+12a12+1b1ε12+12a22+1b2ε22+12μ2ι¯2+ϱ02ϱμ1,where λ=minK¯,βminζ.
By Lemma2, let μ=ζ,ρ=1−ζ,λ=ζ−1,ϕ=∑i=12biβi/2riθ˜i2,andψ=1, then we get(58)∑i=12biβi2riθ˜i2ζ≤∑i=12biβi2riθ˜i2+1−ζζζ/1−ζ.
Then, equation (55) becomes(59)V˙2≤−λVζ+ϱ,where ϱ=1−ζζζ/1−ζ+∑i=12biβi/2riθi2+1/2a12+1/b1ε12+1/2a22+1/b2ε22+1/2μ2ι¯2+ϱ0/2ϱμ1.
Define a positive constant∂ as follows:(60)∂=ϱ1−θ0λ.
Equation (57) can be expressed as(61)Vζzt,θ¯t≤ϱ1−θ0λ.
Fort≥Treach, let(62)Treach=11−θ0λV1−ζz0,θ¯0−ϱ1−θ0λ1−ζ/ζ,with z0=z10,z20T and θ¯0=θ10,θ20T.
Therefore, all the signals are SGPFS.
Then, fort≥Treach, we can obtain that(63)y−qd≤2ϱ1−θ0λ1/2ζ.
Then, the proof is completed.


## 3.1. Adaptive RBFNN Controller Design

Before we begin designing, coordinate transformation is introduced as follows:(24)z1=x1−qd,z2=x2−α1,where α1 is a virtual controller defined as(25)αi=b1N−12ai2ziθ^iℏiZiTℏiZi−12zi−kizi2ζ−1,where ζ>0, ki>0, and ai>0 are design parameters and the ith RBFNN ℏiZi can identify the unknown nonlinear function in the design process. Define the unknown constant θi=ϒi∗2/bi, where θ^i is the estimation of θi and the estimation error is θ˜i=θi−θ^i.Choose the control lawu, the auxiliary controller u¯, and the adaptive law as follows:(26)u=e^u¯,(27)u¯=−12ai2ziθ^iℏiZiTℏiZi−12zi−kizi2ζ−1,(28)θ^.i=ri2ai2zi2ℏiZiTℏiZi−βiθ^i,where ri>0 and σi>0. We assume that θ^it≥0 in this article, e^=1/μ^1 is the estimation of e^=1/μ^1, and e˜=e−e^ is the estimated error. Select e^˙ as(29)e^˙=−ϱznbnu¯−ϱ0e^,where ϱ and ϱ0 are the design constants.Step 1: Lyapunov function candidate will be chosen as follows:(30)V1=12z12+b12r1θ˜12.DifferentiatingV1 yields(31)V1.=z1Nx2+ωt−C−qd−b1r1θ˜1θ^.1.Choose the functionf^1Z1 defined by(32)f^1Z1=−C+ωt−y˙d+k1z12ζ−1,then equation (29) can be rewritten as(33)V˙1=z1f^1+Nx2−b1r1θ˜1θ^.1−k1z12ζ.According to Lemma1, RBFNN ϒ1∗Tℏ1Z1 can be applied to identify the unknown function f^1Z1. For any given ε1>0,(34)f^1Z1=ϒ1∗Tℏ1Z1+ϵ1Z1,ϵ1Z1≤ε1,whereϵ1Z1 is the approximation error. Consequently, let θ1=ϒ1∗2/b1, and one can obtain(35)z1f^1Z1=z1ϒ1∗TΥ1∗Υ1∗ℏ1−z1δ1≤b1z122a12Υ1∗2b1ℏ1Tℏ1+12a12+12b1z12+12b1ε12=b12a12z12θ1ℏ1Tℏ1+12a12+12b1z12+12b1ε12.Combining equations (31) and (33) gives(36)V˙1≤z1b12a12z1θ1ℏ1Tℏ1+Nα1+Nz1z2−b1r1θ˜θ^.1−k1z12ζ+12a12+12b1z12+12b1ε12.According to equation (24) and Assumption 4, it holds that(37)z1Nα1≤​−b12a12z12θ^1ℏ1Tℏ1​−k1b1z12ζ​−12b1z12.Combining (36) with (37) gives(38)V˙1≤b12a12z12θ1ℏ1Tℏ1−b12a12z12θ^1ℏ1Tℏ1−k11+b1z12ζ+Nz1z2−b1r1θ˜θ^.1+12a12+12b1ε12.Choose the adaptation law as(39)θ^.1=r12a12z12ℏ1Z1Tℏ1Z1−βiθ^1.According to equation (37), it holds that(40)b1r1θ˜θ^.1=b12a12z12θ1˜ℏ1Tℏ1−b1β1r1θ˜1θ^1.It is noted that(41)b1β1r1θ˜1θ^1≤−b1β12r1θ˜12+b1β12r1θ12.Therefore, combining equations (38), (40), and (41) can be expressed as(42)V˙1≤ℓ1+Nz1z2,where(43)ℓ1=−k11+b1z12ζ+b1β12r1−θ˜12+θ12+12a12+12b1ε12.Step 2: choose a Lyapunov function candidate as(44)V2=V1+12z22+b22r2θ˜22+μ12ϱe˜2.It is that(45)V˙2=ℓ1+Nz1z2−b2r2θ˜2θ^.2−μ1ϱe˜e^˙+z2fx+gxHu¯+ψxωt−α˙1.Establish the actual controller as(46)u¯=−12a22z2θ^2ℏ2Z2Tℏ2Z2−12z2−k2z22ζ−1.According to equations and Assumption4, one has(47)z2gxHu¯−μϱe˜e^˙≤z2b2u¯−z2b2μ1e˜u¯+z2b2μ2ι−μ1ϱe˜−ϱz2b2u¯−ϱ0e^≤z2b2u¯+z2b2μ2ι+μ1ϱe˜e^≤−12a22z22b2θ^2ℏ2Tℏ2−12b2z22−k2b2z22ζ+12b2z22+12μ2ι¯2+ϱ02ϱμ1,where ι¯=n1/ϑ+ξ.Now, substituting inequalities (47) into (45) results in(48)V˙2≤ℓ1+z2f^2−12a22z22b2θ^2ℏ2Tℏ2−12b2z22−k2b2z22ζ−k2z22ζ+12μ2ι¯2+ϱ02ϱμ1−b2r2θ˜2θ^.2,where f^2=fx+k2z22ζ+Nz1+ψxωt+1/2b22z22−α˙1, which can be estimated by RBFNN ϒ2∗Tℏ2Z2 for any given positive constant ε2. Similarly, let θ2=ϒ2∗2/b2, and one easily obtains(49)z2f^2Z2=z2ϒ2∗TΥ2∗Υ2∗ℏ2−z1δ2≤b2z222a22Υ2∗2b1ℏ2Tℏ2+12a22+12b2z22+12b2ε22=b22a22z22θ2ℏ2Tℏ2+12a22+12b1z22+12b2ε22.Then, equation (46) can be rewritten as(50)V˙2≤ℓ1+b22a22z22θ2ℏ2Tℏ2−12a22z22b2θ^2ℏ2Tℏ2−k21+b2z22ζ+12a22+12b2ε22+12μ2ι¯2+ϱ02ϱμ1−b2r2θ˜2θ^.2.


## 3.2. Stability Analysis

Theorem 1.
Consider system (4). Suppose Assumptions 2–4 hold, then all the signals are SGPFS, under any bounded initial conditions when controller (25) and adaptive law (28) are employed.Proof.
Choose the adaptation law as(51)θ^.2=r22a22z22ℏ1Z2Tℏ1Z2−βiθ^2.
According to equation (49), it holds that(52)b2r2θ˜θ^.2=b22a22z22θ2˜ℏ2Tℏ2−b2β2r2θ˜2θ^2.
It is noted that(53)b2β2r2θ˜2θ^2≤−b2β22r2θ˜22+b2β22r2θ22.
Substituting (52) and (53) into (50), it gives(54)V˙2≤ℓ1+b2β2r2θ˜2θ^2−k2b2z22ζ−k2z22ζ+12μ2ι¯2+ϱ02ϱμ1+12a22+12b2ε22.
Then, we can get(55)V˙2≤−∑i=12ki1+bizi2ζ+b1β12r1−θ˜12+θ12+b2β22r2−θ˜22+θ22+12a12+1b1ε12+12a22+1b2ε22+12μ2ι¯2+ϱ02ϱμ1.
Letkmin=min1≤i≤2ki1+bi, and by Lemma 3, it is deduced that(56)−∑i=12ki1+bizi2ζ≤−kmin∑i=12zi2ζ≤−K¯12∑i=12zi2ζ,where K¯=2ζkmin.
Letβmin=min1≤i≤2βi, and by taking equation (54) into account, inequality (55) can be rewritten as(57)V˙2≤−K¯12∑i=12zi2ζ−βminζ∑i=12bi2riθ˜i2ζ+∑i=12biβi2riθ˜i2ζ+b1β12r1−θ˜12+θ12+b2β22r2−θ˜22+θ22+12a12+1b1ε12+12a22+1b2ε22+12μ2ι¯2+ϱ02ϱμ1≤−λVζ+∑i=12biβi2riθ˜i2ζ+b1β12r1−θ˜12+θ12+b2β22r2−θ˜12+θ12+12a12+1b1ε12+12a22+1b2ε22+12μ2ι¯2+ϱ02ϱμ1,where λ=minK¯,βminζ.
By Lemma2, let μ=ζ,ρ=1−ζ,λ=ζ−1,ϕ=∑i=12biβi/2riθ˜i2,andψ=1, then we get(58)∑i=12biβi2riθ˜i2ζ≤∑i=12biβi2riθ˜i2+1−ζζζ/1−ζ.
Then, equation (55) becomes(59)V˙2≤−λVζ+ϱ,where ϱ=1−ζζζ/1−ζ+∑i=12biβi/2riθi2+1/2a12+1/b1ε12+1/2a22+1/b2ε22+1/2μ2ι¯2+ϱ0/2ϱμ1.
Define a positive constant∂ as follows:(60)∂=ϱ1−θ0λ.
Equation (57) can be expressed as(61)Vζzt,θ¯t≤ϱ1−θ0λ.
Fort≥Treach, let(62)Treach=11−θ0λV1−ζz0,θ¯0−ϱ1−θ0λ1−ζ/ζ,with z0=z10,z20T and θ¯0=θ10,θ20T.
Therefore, all the signals are SGPFS.
Then, fort≥Treach, we can obtain that(63)y−qd≤2ϱ1−θ0λ1/2ζ.
Then, the proof is completed.


## 4. Simulation Example

In this section, to certify the feasibility of the strategy presented in this work, a simulation example is given by MATLAB.Select the system parameters and external disturbance in system (4) as(64)N=1,R=6s,C=2.5packets/s,a1=0.3,a2=2,k1=k2=0.1,r1=r2=20,β1=0.08,β2=0.2,ω=0.5e−0.5tsin6t.Define the hysteresisHu as follows:(65)Hu=μ1u+μ2ι,where μ1=10 and μ2=0.5. ι is selected as equation (8). The function fι,u˙ is selected as equation (7).The initial values of the state areϑ=2, ξ=1, and n=3; y0=0.5.The results of simulation are shown in Figures1–6. Figure 1 shows the tracking error of qt and qd. It is clear that the queue can track the desired queue within the allowable error. Figure 2 introduces the hysteresis output Hu. Besides, Figures 3 and 4 show the trajectory of the adaptive law, from which we can obtain that all the adaptive laws are SGPFS. Figure 5 shows the trajectory of the rate rt. As a result, the proposed strategy is effective.Figure 1
Tracking errorqt−qd.Figure 2
The hysteresis outputHu.Figure 3
The trajectory of the adaptive law 1.Figure 4
The trajectory of the adaptive law 2.Figure 5
The sending ratert.Figure 6
The comparison results with the tracking error.In addition, in order to illustrate the advantages of this algorithm, the simulation comparisons are made between the method in this paper and the random early detection (RED) algorithm. The comparison result is shown in Figure6, in which the preset properties with respect to qt−qd are obtained. Moreover, it should be pointed out that the maximum overshoot of qt−qd is less than 0.8. Further, it is easy to observe from Figure 6 that the smaller overshoot and the less chattering are achieved comparing with RED algorithm. As a result, the proposed method has the better performances.


## 5. Conclusion

In this manuscript, an adaptive FT control is considered for TCP/AQM networks. The finite-time controller by using the backstepping technique and the adaptive neural control method ensures that the queue length tracks the desired queue length and the tracking error converges to the prescribed area. Besides, the controller can not only reduce the influence of the disturbance, but also shorten the impression of uncertainty. Finally, an example is offered to verify the superiority and effectiveness.



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*Source: 4138390-2020-07-27.xml*