CN103354526A - Fractional-order global sliding-mode Internet congestion control method - Google Patents

Abstract

The invention relates to a fractional-order global sliding-mode Internet congestion control method, belonging to the field of Internet congestion control. To solve congestion problems existing in the Internet, the method utilizes strong robustness of a global sliding mode and good memory of fractional order to improve the congestion control effect. A parameter perturbation item is introduced into a network congestion control model to represent change of practical network parameters. A gradually-stable global sliding-mode surface with a fractional order item is designed to effectively inhibit influence of time-varying parameters to a system. A designed fractional-order sliding-mode controller can effectively reduce buffeting phenomena, so that operation of the system is similar to a sliding-mode movement on the sliding-mode surface. According to the method of the invention, congestion of the network is effectively inhibited, and service quality of the network is improved.

Description

Fractional order global sliding mode internet congestion control method

Technical field

The invention belongs to congestion control field, the Internet.

Background technology

The fast development of the Internet in the world distance between the different geographical that furthered makes the quick interchange between the different geographical become possibility.But because the Internet is comprised of each different computing terminal, router and the communication line of different geographical, therefore, isomerism is its intrinsic characteristic, thereby has caused the area information that has fast alternately, and the area information that has is slow alternately.When the scale development of Internet time to a certain degree, congestion phenomenon just almost invariably often occurs.Be reflected in it the user, it is very slow that maximum feeling is exactly that networking speed becomes, and sometimes even can't surf the Net, this online that has greatly reduced the user is experienced.Because Internet's is huge, it is also unrealistic merely to increase the hardware input, therefore at the network node place, mainly is to introduce the congestion control module in router, wherein moves jamming control method, is the effective way that addresses this problem.

In recent years, network congestion control has become a hot research field, some preferably jamming control methods in succession occurred.At document Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED (Proceedings of ACM/SIGCOMM, 2000:151-160), the author has set up the network congestion control model of differential equation form first to the TCP network, thereby the basis of having established model for follow-up research can more easily be applied in the Internet various advanced persons' control theory.At document Adaptive active queue management controller for TCP communication networks using PSO-RBF models (Neural Computing﹠amp; Applications, 2013,22 (5): 933-945), the author has proposed a kind of linear R BF congestion controller, and has obtained the output weight of this controller with particle swarm optimization algorithm, makes the entire system error minimize.

Yet the Internet is a system that is full of various uncertain factors, and this just requires controling appliance to have extremely strong robustness that good congestion control effect can be arranged.Sliding mode in the sliding formwork control has consistency, and these characteristics make it be fit to very much the congestion control of the Internet.In recent years, some have occurred and sliding formwork control has been applied to the method in this field.At document Linear quadratic optimal discrete-time sliding-mode controller for connection-oriented communication networks (IEEE Transactions on Industrial Electronics, 2008,55 (11): 4013-4021), the author has proposed a sliding mode controller with Linear-Quadratic Problem, and has proved the stability of closed-loop system.At document Discrete time sliding mode flow controller for multi-source single-bottleneck connection-oriented communication networks (Journal of Vibration and Control, 2009,15 (11): 1745-1760), the author is discrete system with noisy bandwidth varying with network modelling, and the method that proposes can guarantee system's closed-loop stabilization and finite time convergence.The control of the nonlinear adaptive sliding formwork of document TCP network (Northeastern University's journal (natural science edition), 2012,33 (11): 1521-1524), author designed two kinds of adaptive sliding mode controllers, its performance is better than the PI controller.But also all there are some shortcomings separately in above-mentioned document, if any the chattering phenomenon of document comparatively obvious, the transient characterisitics of the document that has are not ideal enough etc.

New fractional-order system is to be based upon fractional calculus and the theoretic model system of Fractional Differential Equation, fractional calculus has been considered the Global Information of system with the form of weighting, has good Memorability, can describe more accurately the dynamic response of real system, expand the descriptive power of integer rank calculus.In recent years, the research of new fractional-order system has obtained people's attention, at electric system, the viscoelasticity system, obtained certain application in biosystem and the robot system, document Fractional fuzzy adaptive sliding-mode control of a2-DOF direct-drive robot arm (IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics for example, 2008,38 (6): 1561-1570), document Sensorless vector control of PMSM using sliding mode observer and fractional-order phase-locked loop (Proceedings of the31st Chinese Control Conference, 2012:4513-4518) etc.If new fractional-order system is applied in the network congestion control, will obtain better congestion control effect, but regrettably, up to the present, the research of this respect but rarely has report.

Summary of the invention

In order to solve better the congestion problems that exists in the Internet, the object of the invention is to propose a kind of fractional order global sliding mode jamming control method, fractional order global sliding mode control theory is incorporated in the congestion control of the Internet.The invention is characterized in that the method contains has the following steps:

(A) initial value of setting each network parameter is separately nominal value, specifically comprise: the initial value of the TCP linking number N (t) of network is made as N, the initial value of the trunk link bandwidth C (t) of network is made as C, the initial value of the round-trip delay R (t) of packet is made as R ₀And, the desired value of the window size W (t) of network is set as W ₀, the desired value of queue length in router q (t) is set as q ₀, and according to p ₀=2N ²/ (R ₀ ²C ²) calculate grouping and abandon/the desired value p of marking probability p (t) ₀, be greater than zero constant with the parameter k in the controller and the equal value of ε.

(B) begin the packet that enters into congested router is sampled, obtain the initial value of W (t) and q (t), deduct respectively separately desired value with initial value, obtain the initial value of system mode.

(C) every 10ms once sampling network parameter N (t), the numerical value of C (t) and R (t).

(D) set up suc as formula the control of the network congestion shown in (1) and the formula (2) model according to above-mentioned network parameter.

$\mathrm{\δ}\stackrel{\·}{W}\left(t\right)=-\frac{2N}{{R_0}^{2}C}\mathrm{\δW}\left(t\right)-\frac{R_0{C}^2}{{2N}^2}\mathrm{\δp}\left(t\right)---\left(\text{1}\right)$

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{q}\left(t\right)=\frac{\mathrm{<figure-callout\; id="0"\; label="N\; R"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">N</figure-callout>}}{R_{}}\mathrm{\&delta;W}\left(t\right)-\frac{1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}\mathrm{\&delta;q}\left(t\right)---\left(2\right)$

Wherein: δ W (t)=W (t)-W ₀, δ q (t)=q (t)-q ₀, δ p (t)=p (t)-p ₀.

(E) make x ₁(t)=and δ q (t), x ₂(t)=and δ W (t), u (t)=δ p (t) sets up the system state space equation.

(F) the variation equivalence with network parameter is the perturbation of system parameters, obtains the network congestion control model with the Parameter Perturbation item.

(G) determine the variation boundary μ of Parameter Perturbation item according to the variation of network parameter and nominal value separately _AAnd μ _B

(H) set fractional order global sliding mode face S (t) and have following form:

$S\left(t\right)=-\mathrm{Qx}\left(t\right)+D_t^{\mathrm{\&alpha;}}{}_{0}x\left(t\right)-(-\mathrm{Qx}\left(0\right)+\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">D</figure-callout>}_0^{\mathrm{\&alpha;}}{}_{0}x\left(0\right))e^{-t}$

Wherein:

Be that 0＜α＜1 is the order of fractional order according to the fractional order differential operator of G-L definition, the characteristic value of matrix Q is λ ₁And λ ₂, and satisfy | arg (λ ₁) | the α pi/2, | arg (λ ₂) | the α pi/2.

(I) set fractional order global sliding mode controller u (t) and have following form:

$u\left(t\right)={\left(\mathrm{QB}\right)}^{-1}[D_t^{\mathrm{\&alpha;}+1}{}_{0}x\left(t\right)-d\left((-\mathrm{Qx}\left(0\right)+\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">D</figure-callout>}_0^{\mathrm{\&alpha;}}{}_{0}x\left(0\right))e^{-t}\right)/\mathrm{dt}-\mathrm{Qax}\left(t\right)+$ $\left({\mathrm{\&mu;}}_A\right|\left|\mathrm{Qx}\left(t\right)\right||+{\mathrm{\&mu;}}_B|\left|Q\right|\left|\right)\mathrm{signS}\left(t\right)+\mathrm{kS}\left(t\right)+\mathrm{\&epsiv;}(1-e^{-\left|\right|x\left(t\right)\left|\right|})\mathrm{signS}\left(t\right)]$

Wherein: matrix A and B are the coefficient matrix of the system state space equation that obtains in the step (E), and coefficient k and ε are the constant greater than zero.To obtain the concrete numerical value of u (t) according to concrete network parameter.If u (t)＜-p ₀, then need with u (t) be modified to u (t)=-p ₀If u (t)〉1-p ₀, then need u (t) is modified to u (t)=1-p ₀In other situation, the numerical value of u (t) remains unchanged.

(J) insert congestion marking at router take p (t) as probability in response to the respond packet of transmitting terminal, so that transmitting terminal is according to correspondingly adjusting transmission rate with what of congestion marking number of packet, thus the increasing the weight of of the generation of avoid congestion or degree.

Compare with existing method, the present invention has following 2 advantages:

(1) strong robustness.This method is the global sliding mode method, does not have approach procedure, and controller can effectively reduce the chattering phenomenon of system, and the overall situation of system motion all can be similar to thinks sliding mode, therefore, and the strong robustness of system.

(2) fractional order global sliding mode control method is applied in the network congestion control.The fractional calculus operator has good Memorability, and the substantive characteristics of descriptive system combines fractional order and global sliding mode control better, thereby has improved further the congestion control effect of network.

In a word, the present invention is applied to fractional order and global sliding mode control theory in the congestion control of the Internet.At first the variation equivalence with network parameter is the perturbation of system parameters, has then designed the asymptotically stable global sliding mode face with the fractional order item, has removed the approach procedure of system, and system motion is in the sliding-mode surface at the very start, has improved the robustness of system.And the chattering phenomenon that designed controller can the establishment system makes the vibration of the queue length in the router obtain good control.The present invention has suppressed the congestion phenomenon of network effectively, has improved user's online and has experienced.

Description of drawings

The reponse system model of Fig. 1 network congestion control.

Control effect when Fig. 2 network parameter is nominal value.

Control effect during Fig. 3 N (t) change at random.

Control effect during Fig. 4 C (t) change at random.

Control effect during Fig. 5 R (t) change at random.

Fig. 6 N (t), the control effect during the equal change at random of C (t) and R (t).

Embodiment

Embodiment 1

Existing document has provided the Internet Congestion Control Model as follows:

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{W}\left(t\right)=-\frac{2N}{{R_0}^{2}C}\mathrm{\&delta;W}\left(t\right)-\frac{R_0{C}^2}{{2N}^2}\mathrm{\&delta;p}\left(t\right)---\left(\text{1}\right)$

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{q}\left(t\right)=\frac{\mathrm{<figure-callout\; id="0"\; label="N\; R"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">N</figure-callout>}}{R_{}}\mathrm{\&delta;W}\left(t\right)-\frac{1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}\mathrm{\&delta;q}\left(t\right)---\left(2\right)$

Wherein: N, C and R ₀Be respectively N (t), the nominal value of C (t) and R (t), N (t) are the TCP linking number of network, and C (t) is the trunk link bandwidth of network, and R (t) is the round-trip delay of packet, δ W (t)=W (t)-W ₀, W (t) is the currency of TCP network window, W ₀Be the desired value of TCP network window, δ q (t)=q (t)-q ₀, q (t) is the currency of queue length in the router, q ₀Be the desired value of queue length in the router, δ p (t)=p (t)-p ₀, 0≤p (t)≤1 abandons/marking probability for packet, and W is arranged ₀ ²p ₀=2, p ₀=2N ²/ (R ₀ ²C ²).

Make x ₁(t)=and δ q (t), x ₂(t)=and δ W (t), u (t)=δ p (t), then formula (1) and formula (2) can turn to

$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)---\left(3\right)$

Wherein: $x\left(t\right)=\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right],$ $A=\left[\begin{array}{cc}-\frac{1}{R_0}& \frac{N}{R_0}\\ 0& -\frac{2N}{{R_0}^{2}C}\end{array}\right],$ $B=\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left[\begin{array}{c}0\\ -\frac{R_0{C}^2}{{2N}^2}\end{array}\right],$ -p ₀≤u(t)≤1-p ₀.

Introduce the Parameter Perturbation item in formula (3), in order to the variation of equivalent network parameters, thereby formula (3) can turn to

$\stackrel{\&CenterDot;}{x}\left(t\right)=(A+\mathrm{\&Delta;A}\left(t\right))x\left(t\right)+(B+\mathrm{\&Delta;B}\left(t\right))u\left(t\right)---\left(4\right)$

Wherein: Δ A (t)=A (t)-A and Δ B (t)=B (t)-B is the Parameter Perturbation item in the system.

Because network parameter N (t), the variation of C (t) and R (t) is limited, so Parameter Perturbation item Δ A (t) and Δ B (t) be norm-bounded, and boundary might as well be used respectively μ _AAnd μ _BExpression, the size of the two is determined by following formula:

μ _A=max|‖A(t)‖ ₁-‖A‖ ₁|,μ _B=max|‖B(t)‖ ₁-‖B‖ ₁| (5)

Fractional order global sliding mode face is set as following form:

$S\left(t\right)=-\mathrm{Qx}\left(t\right)+D_t^{\mathrm{\&alpha;}}{}_{0}x\left(t\right)-(-\mathrm{Qx}\left(0\right)+\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">D</figure-callout>}_0^{\mathrm{\&alpha;}}{}_{0}x\left(0\right))e^{-t}---\left(6\right)$

Wherein:

Be that 0＜α＜1 is the order of fractional order according to the fractional order differential operator of G-L definition, the characteristic value of matrix Q is λ ₁And λ ₂, and satisfy | arg (λ ₁) | the α pi/2, | arg (λ ₂) | the α pi/2, utilize existing document to know that such design can guarantee the sliding mode Asymptotic Stability on the sliding-mode surface.

Here get α=0.25, $Q=\left[\begin{array}{cc}1& -2\\ 5& 3\end{array}\right],$ So λ ₁=2+3i and λ ₂=2-3i, can satisfy | arg (λ ₁) | π/8 Hes | arg (λ ₂) | the requirement of π/8.

Fractional order global sliding mode controller is set as following form:

$u\left(t\right)={\left(\mathrm{QB}\right)}^{-1}[D_t^{\mathrm{\&alpha;}+1}{}_{0}x\left(t\right)-d\left((-\mathrm{Qx}\left(0\right)+\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="HDA00003483528000013.png,HDA00003483528000014.png"\; state="\{\{state\}\}">D</figure-callout>}_0^{\mathrm{\&alpha;}}{}_{0}x\left(0\right))e^{-t}\right)/\mathrm{dt}-\mathrm{Qax}\left(t\right)+$

$\left({\mathrm{\&mu;}}_A\right|\left|\mathrm{Qx}\left(t\right)\right||+{\mathrm{\&mu;}}_B|\left|Q\right|\left|\right)\mathrm{signS}\left(t\right)+\mathrm{kS}\left(t\right)+\mathrm{\&epsiv;}(1-e^{-\left|\right|x\left(t\right)\left|\right|})\mathrm{signS}\left(t\right)]---\left(7\right)$

Wherein: coefficient k and ε are the constant greater than zero, get k=12 here, ε=4.

By formula (4), formula (6) and formula (7) know that such controller can satisfy the following Reaching Law in the existing document:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{S}\left(t\right)\&le;-\mathrm{kS}\left(t\right)-\mathrm{\&epsiv;}(1-e^{-\left|\right|x\left(t\right)\left|\right|})\mathrm{signS}\left(t\right),S\left(t\right)>0\\ \stackrel{\&CenterDot;}{S}\left(t\right)\&GreaterEqual;-\mathrm{kS}\left(t\right)-\mathrm{\&epsiv;}(1-e^{-\left|\right|x\left(t\right)\left|\right|})\mathrm{signS}\left(t\right),S\left(t\right)<0\end{array}\right.---\left(8\right)$

The reponse system model of the network congestion control that is comprised of controller and controlled device as shown in Figure 1.Known that by existing document the controller that satisfies Reaching Law formula (8) can effectively reduce the chattering phenomenon of sliding mode system, namely formula (7) can the establishment router in the vibration of queue length, improve the effect of congestion control.

Each network parameter that sampling instant is obtained is updated in the formula (7), gets final product the concrete numerical value of controlled device u (t), when the u that obtains (t) satisfies-p ₀≤ u (t)≤1-p ₀, then u (t) remains unchanged.Otherwise, if u (t)＜-p ₀, then u (t) is modified to u (t)=-p ₀If u (t)〉1-p ₀, then u (t) is modified to u (t)=1-p ₀

The nominal value of hypothetical network Parameter N (t) is 2000, the nominal value of C (t) is 50Mbps, the nominal value of R (t) is 110ms, the desired value of TCP network window is 30packets, the initial value of TCP network window is 20packets, the largest buffered of router is 700packets, and the queue length of expectation is 400packets, and initial queue length is 300packets.

According to above-mentioned network parameter and formula (1), formula (2) and formula (3)

$A=\left[\begin{array}{cc}-9.0909& 1.8181\&times;{10}^4\\ 0& -6.6116\&times;{10}^{-3}\end{array}\right],$ $B=\left[\begin{array}{c}0\\ -3.4375\&times;{10}^7\end{array}\right],$ p ₀=2.6446×10 ^-7.

Robustness for verification system, suppose N (t) change at random between 1500 to 2500, C (t) at 35Mbps to change at random between the 65Mbps, R (t) at 80ms to change at random between the 140ms, the variation of these network parameters is equivalent to Parameter Perturbation concerning system, be equivalent in other words add interference, get μ according to formula (5) _A=1.3068 * 10 ⁴, μ _B=9.7069 * 10 ⁷Concrete numerical value with can controlled device in the above parameter substitution formula (7) changing with the variation of network parameter has namely obtained concrete fractional order global sliding mode internet congestion control method.

Fractional order global sliding mode internet congestion control method of the present invention may be summarized to be following steps:

(A) initial value of setting each network parameter is separately nominal value, specifically comprise: the initial value of the TCP linking number N (t) of network is made as N, the initial value of the trunk link bandwidth C (t) of network is made as C, and the initial value of the round-trip delay R (t) of packet is made as R ₀And, the desired value of the window size W (t) of network is set as W ₀, the desired value of queue length in router q (t) is set as q ₀, and according to p ₀=2N ²/ (R ₀ ²C ²) calculate grouping and abandon/the desired value p of marking probability p (t) ₀, with the parameter k in the controller and ε respectively value be 12 and 4.

(B) begin the packet that enters into congested router is sampled, obtain the initial value of W (t) and q (t), deduct respectively separately desired value with initial value, obtain the initial value x (0) of system mode.

(C) every 10ms once sampling network parameter N (t), the numerical value of C (t) and R (t).

(D) set up suc as formula the control of the network congestion shown in (1) and the formula (2) model according to above-mentioned network parameter.

(E) set up the system state space equation according to formula (3).

(F) the variation equivalence with network parameter is the perturbation of system parameters, obtains suc as formula the network congestion control model with the Parameter Perturbation item shown in (4).

(G) the variation boundary of determining the Parameter Perturbation item according to variation and the formula (5) of network parameter.

(H) obtain the expression of fractional order global sliding mode face S (t) according to above-mentioned network parameter and formula (6).

(I) obtain the concrete numerical value of fractional order global sliding mode controller u (t) according to above-mentioned network parameter and formula (7).If u (t)＜-p ₀, then u (t) is modified to u (t)=-p ₀If u (t)〉1-p ₀, then u (t) is modified to u (t)=1-p ₀In other situation, the numerical value of u (t) remains unchanged.

(J) insert congestion marking at router take p (t) as probability in response to the respond packet of transmitting terminal, so that transmitting terminal is according to correspondingly adjusting transmission rate with what of congestion marking number of packet, thus the increasing the weight of of the generation of avoid congestion or degree.

In order to verify the congestion control effect, we have carried out emulation with the NS2 Network Simulation Software to the method.Artificial network adopts the network topology structure of dumbbell type.At first, make N (t), C (t) and R (t) are above-mentioned nominal value separately, and it is constant to keep nominal value, and the simulation result that obtains as shown in Figure 2.As seen from Figure 2: this method makes the control target, and namely the queue length in the router quickly converges on desired value, and steady-state error is very little, and this is because the method can effectively reduce the result of the chattering phenomenon of system.Secondly, checking is when the equal robust performance during change at random in previously described scope of each network parameter.Specific practice is: (1) makes N (t) change at random between 1500 to 2500 after system moves 20 seconds, then after system moves 80 seconds, makes N (t) return to nominal value again, and the simulation result that obtains as shown in Figure 3.(2) after system operation 20 seconds, make C (t) at 35Mbps to change at random between the 65Mbps, then after system moves 80 seconds, make C (t) return to again nominal value, the simulation result that obtains as shown in Figure 4.(3) after system operation 20 seconds, make R (t) at 80ms to change at random between the 140ms, then after system moves 80 seconds, make R (t) return to again nominal value, the simulation result that obtains as shown in Figure 5.(4) after system moves 20 seconds, make N (t), C (t) and R (t) be change at random in previously described scope all, then after system moves 80 seconds, makes them return to nominal value again, and the simulation result that obtains as shown in Figure 6.From Fig. 3 to Fig. 6, can find out: no matter be the independent change at random of each network parameter, or the equal change at random of all-network parameter, system all can make the queue length in the router be stabilized near the desired value through after the of short duration adjustment rapidly, has shown good congestion control effect.Why having such effect is because the method has adopted global sliding mode control, and has utilized the good characteristics of new fractional-order system Memorability, and has also considered the impact of uncertain factor in system model, the coefficient result of these measures.Robustness well means to make always certain spatial cache more than needed in the router, be conducive to like this tackle the impact of bursty traffic, and then the generation of avoid congestion.

Claims (1)

1. fractional order global sliding mode internet congestion control method is characterized in that comprising following steps:

(A) initial value of setting each network parameter is separately nominal value, specifically comprise: the initial value of the TCP linking number N (t) of network is made as N, the initial value of the trunk link bandwidth C (t) of network is made as C, the initial value of the round-trip delay R (t) of packet is made as R ₀And, the desired value of the window size W (t) of network is set as W ₀, the desired value of queue length in router q (t) is set as q ₀, and according to p ₀=2N ²/ (R ₀ ²C ²) calculate grouping and abandon/the desired value p of marking probability p (t) ₀, be greater than zero constant with the parameter k in the controller and the equal value of ε;

(B) begin the packet that enters into congested router is sampled, obtain the initial value of W (t) and q (t), deduct respectively separately desired value with initial value, obtain the initial value of system mode;

(C) every 10ms once sampling network parameter N (t), the numerical value of C (t) and R (t);

(D) set up suc as formula the control of the network congestion shown in (1) and the formula (2) model according to above-mentioned network parameter;

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{W}\left(t\right)=-\frac{2N}{{R_0}^{2}C}\mathrm{\&delta;W}\left(t\right)-\frac{R_0{C}^2}{{2N}^2}\mathrm{\&delta;p}\left(t\right)---\left(\text{1}\right)$

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{q}\left(t\right)=\frac{N}{R_0}\mathrm{\&delta;W}\left(t\right)-\frac{1}{R_0}\mathrm{\&delta;q}\left(t\right)---\left(2\right)$

Wherein: δ W (t)=W (t)-W ₀, δ q (t)=q (t)-q ₀, δ p (t)=p (t)-p ₀

(E) make x ₁(t)=and δ q (t), x ₂(t)=and δ W (t), u (t)=δ p (t) sets up the system state space equation;

(F) the variation equivalence with network parameter is the perturbation of system parameters, obtains the network congestion control model with the Parameter Perturbation item;

(G) determine the variation boundary μ of Parameter Perturbation item according to the variation of network parameter and nominal value separately _AAnd μ _B

(H) set fractional order global sliding mode face S (t) and have following form:

$S\left(t\right)=-\mathrm{Qx}\left(t\right)+D_t^{\mathrm{\&alpha;}}{}_{0}x\left(t\right)-(-\mathrm{Qx}\left(0\right)+D_0^{\mathrm{\&alpha;}}{}_{0}x\left(0\right))e^{-t}$

Wherein: Be that 0＜α＜1 is the order of fractional order according to the fractional order differential operator of G-L definition, the characteristic value of matrix Q is λ ₁And λ ₂, and satisfy | arg (λ ₁) | the α pi/2, | arg (λ ₂) | the α pi/2;

(I) set fractional order global sliding mode controller u (t) and have following form:

$u\left(t\right)={\left(\mathrm{QB}\right)}^{-1}[D_t^{\mathrm{\&alpha;}+1}{}_{0}x\left(t\right)-d\left((-\mathrm{Qx}\left(0\right)+D_0^{\mathrm{\&alpha;}}{}_{0}x\left(0\right))e^{-t}\right)/\mathrm{dt}-\mathrm{QAx}\left(t\right)+$

$\left({\mathrm{\&mu;}}_A\right|\left|\mathrm{Qx}\left(t\right)\right||+{\mathrm{\&mu;}}_B|\left|Q\right|\left|\right)\mathrm{signS}\left(t\right)+\mathrm{kS}\left(t\right)+\mathrm{\&epsiv;}(1-e^{-\left|\right|x\left(t\right)\left|\right|})\mathrm{signS}\left(t\right)]$

Wherein: matrix A and B are the coefficient matrix of the system state space equation that obtains in the step (E), and coefficient k and ε are the constant greater than zero; To obtain the concrete numerical value of u (t) according to concrete network parameter; If u (t)＜-p ₀, then need with u (t) be modified to u (t)=-p ₀If u (t)〉1-p ₀, then need u (t) is modified to u (t)=1-p ₀In other situation, the numerical value of u (t) remains unchanged;

(J) insert congestion marking at router take p (t) as probability in response to the respond packet of transmitting terminal, so that transmitting terminal is according to correspondingly adjusting transmission rate with what of congestion marking number of packet, thus the increasing the weight of of the generation of avoid congestion or degree.

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