International Journal of Analysis and Applications Volume 16, Number 5 (2018), 654-672 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-654 ROBUST STABILITY AND DESIGN OF STATE FEEDBACK CONTROLLER FOR STRAIGHTFORWARD ACTIVE QUEUE MANAGEMENT MEHRDAD NOZOHOUR YAZDI AND ALI DELAVARKHALAFI∗ Department of Mathematics, Yazd University, Yazd, Iran ∗Corresponding author: delavarkh@yazd.ac.ir Abstract. The straightforward active queue management (AQM), which is based on the prediction of arrival rate is investigated by means of state-space approach. We formulate the feedback control design problem for linearized system of additive increase multiplicative decrease (AIMD) dynamic models as state-space model. Then the Lyapunov-Krasovskii method is provided to achieve the robust stability and sufficient stabilization condition and afterwards the term of linear inequality matrix (LMI) is used to show the results. We present the simulation results and show the superiority of our proposed method to other control mechanisms. 1. Introduction Nowadays, computer networks develop rapidly and the growth of the amount of data communicated in computer networks become an important issue in this field. Since the available capacity of the resource is usually less than the demand, congestion happens which is the main concerns to deal with in computer networks. Two effective strategies to control congestion are Transmission Control Protocol (TCP) and Active Queue Management (AQM) which they control the congestion at the end and at the hosts, respectively. TCP recognize the congestion by receiving acknowledgements and moderate the TCP window sizes of senders(ref in congestion control paper). Among the categories in control systems, networked control systems (NCS) are a category in which sensors, actuators, and controllers are connected though a network. Different forms Received 2017-10-29; accepted 2018-01-10; published 2018-09-05. 2010 Mathematics Subject Classification. 93B05, 93E15. Key words and phrases. network congestion control; active queue management(aqm); packet arrival rate; prediction; asymp- totical stability. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 654 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-654 Int. J. Anal. Appl. 16 (5) (2018) 655 and investigations in NCS have been exposed in [11]. For designing the control systems, parameter effects such as delay [14] and [10], quality of service (QoS), and co-design of plant controller [7], [18], [26] and [30] are considered. The limitation on the amount of delay is assumed and there is a lot of attention on packet lost [16]. Dai et al. [8] improved the quality of service (QoS) and propagate the control performance of the system. In TCP/AQM, Dynamical models for average TCP window size and the queue size by including the TCP networks and AQM strategies , are provided [12], [13] and [20]. In TCP/AQM models after detecting the congestion, they convey the information to end host, in which the routers drop or mark, when explicit congestion notification (ECN) is activated, packets and they do all these things before the buffer over- flows. Also, end hosts after redacting their congestion window size, reconcile the sending rates. Since packet-dropping causes flow-synchronization and performance depreciation, RED scheme was represented to allow to routers to facilitate TCP ’s management of network performance [9]. Hollot et al. [13] derived proportional and proportional-integral control for AQM schemes in which, the dynamical model developed by Misra et al. [20]. The AQM algorithms based on control theory is complicated to implement and some times parameter setting effects the system design [1] and [23]. To overcome with this, AQM algorithm with simple implementation based on prediction of arrival rate was presented in [27] and this prediction was derived from the analysis of the network congestion control. Since these dynamical models are nonlinear, not only they are linearized at the operating point, but also this linearization is depend on a time delay which play an important role in models and their stability. Time delay can be an uncertainties for the system and robust controller such as H∞ could be designed for them [21]. In order to rectify the delays and the impact on system performance, the state-space models developed to formulate the control design problem [4]. Wang et al. [24] use the state-space model to derive model predictive controller and use the motion equation to linearized the system. The queue length in data buffer was predicted there and then packets at router were dropped. When the linear system obtained, stability analysis and stabilization of linear time delay systems must be considered. The stability of linear system may be required to consider the poles of closed-loop transfer function. The routh-Hurwitz stability of a linear system, provides the stability by finding the roots of characteristic equation of the system [5], [6] and [15]. The Nyquist criterion is a graphical method and deals with the loop gain transfer function, i.e., the open loop transfer function. Hollot et al. [13] find the stability of PI and PID control and also represented that RED’s queue-averaging is not beneficial by Nyquist stability. Nyquist plot specifies the factor in which the system magnitude and makes the system marginally stable [15]. That factor is called gain margin. Phase margin is obtained by phase shift which determined by Nyquist plot or bode plot [12]. Stability theory influences the system theory and engineering and also the stability implies the behavior of a system over a long time period. The asymptotic behavior of a state of the system near the Int. J. Anal. Appl. 16 (5) (2018) 656 steady-state solution like equilibrium points or operating points, helps to derive the stability of systems. The stability of equilibrium points, characterized in the sense of Lyapunov. Stability and asymptotically stability of system, were defined in [15]. The contribution of this investigation is to model state-space of the active queue management in which the drop probability policy is outspoken. The reason to go through to this designing is finding the asymptotical stability of the system which is the significant basis of the affectivity of the system. Lyapunov method is used to locate the stability conditions of an equilibrium point without solving the state equation. The remain of this paper is organized into description of the straightforward active queue management in section 2. The control model of this system and stability condition for proposed method are derived in section 3. In section 4, the illustrative examples are provided to represented the effectiveness of the method. Conclusion and suggestion of future work are located in section 5. 2. Prediction of the change of packet arrival rate In order to formulate the system we first provide some relations. The system is a single congested router with a capacity C. Let N, TCP flows labeled i = 1, ...,N, Wi(t), Vi(t) and Ri(t) represent the congestion window size, packet sending rate and Round Trip Time(RTT)of flow TCPi(i = 1, . . . ,N) at time t > 0, respectively. The packet arrival rate at time t > 0 is λ(t), then Vi(t) = Wi(t) Ri(t) , (2.1) λ(t) = n∑ i=1 Vi(t), Ri(t) = ri + q(t) C , (2.2) where ri is the Propagation of RRT that we called RTPT of TCPi, q(t) is the queue length, and the queueing delay of models is q(t) C . Additive Increase and Multiplicative Decrease (AIMD) is the famous strategy is used here and the full de- scription of this strategy can be found in [19]. The expectation of the increment of the arrival rate is as follows: ∆λ(t) = 1 Wi(t)Ri(t) (1 −p(t− τ)) − Vi(t) 2 p(t− τ). (2.3) The moment that the packets are dropped or acknowledged to the moment that host receives the informa- tion. Here in this system Wi(t), Vi(t) and Ri(t) are unknown variables and also we assume that they have same fixed propagation delay Ri(t) and we approximated to their expectations. The other assumptions here, are the desired value of queue occupation which is shown as qref and the average value of RTPT of all TCP sessions which is r. Hence the RRT could be derived as: Ri(t) ≈ r + qref C = R, (2.4) Int. J. Anal. Appl. 16 (5) (2018) 657 where R is a RRT approximation. Afterward the expectation of the sending rate Vi(t) and window size Wi(t) will be expressed as follows whilst Vi(t) λ(t) is proportion of the packets that are generated by TCPi, V (t) = n∑ i=1 Vi(t) Vi(t) λ(t) ≥ ( ∑n i=1 Vi(t)) 2 Nλ(t) = λ(t) N , (2.5) W(t) = n∑ i=1 Wi(t) Vi(t) λ(t) ≥ ( ∑n i=1 Vi(t)) 2R Nλ(t) = λ(t)R N . (2.6) Since the system is based on packet dropping probability, so we assume it updated at every time interval and represented as p(t). Also the number of arriving packets at the router is m(t). Now by combing the equations (2.4-2.6), the expectation of arrival at the router is obtained below, is the objective of analysis which is provided by ( [19] and [27]) and they shown that there is a mismatch between the operating value of the queue length and desired one by their method, actually it’s smaller than the desired one. ∆λ(t + RRT) = ∑m(t) N(1−p(t)) λ(t+RRT)R2 i (t+RRT) − λ(t+RRT)p(t)m(t) 2N ≈ N(1−p(t)) λ(t+RRT)R2 i − λ(t+RRT)p(t)m(t) 2N , (2.7) Xu and Sun [19] proposed the statistical method which is related to this dropping probability and they named it straightforward active queue management (SFAQM) . The arrival rate when the congestion information arrive at the hosts is λ(t + RRT) and the desired arrival rate is λref (t + RRT). So in order to reach to the desired queue length qref we obtain: λ(t + RRT) + ∆λ(t + RRT) = λref (t + RRT), (2.8) λref (t + RRT) = C + (qref−q(t)) α . (2.9) Here, α is a parameter to control queue length to desired value. Hence, p(t) = λ(t + RRT) + m(t)N λ(t+RRT)R2 −λref (t + RRT) ( λ(t+RRT) 2N + N λ(t+RRT)R2 )m(t) (2.10) By counting the number of arriving packets m(t) to estimate the arrival rate λ(t) will be expressed as: λ(t) = m(t) δ , (2.11) Int. J. Anal. Appl. 16 (5) (2018) 658 Figure 1. The Flowchart of SFAQM [27]. where δ is the length of the sampling period. Substituting (2.11) in (2.10) we can conclude: p(t) = m(t + RRT) + m(t)Nδ2 m(t+RRT)R2 −mref (t + RRT) ( m(t+RRT) 2N + Nδ 2 m(t+RRT)R2 )m(t) , (2.12) where mref (t + RRT) shows as follows: mref (t + RRT) = Cδ + (qref −q(t))δ α . (2.13) The exponential weighted moving average (EWMA) was used to predict m(t) by noticing that m(t + RRT) is predicted as a value between m(t) and mref (t + RRT), we have: m(t) = (1 −ω1)m(t− δ) + ω1m0, (2.14) m(t + RRT) = (1 −ω2)m(t) + ω2mref (t + RRT), (2.15) where • ω1, ω2: weight parameter, • m0: number of arriving packets during the previous interval, • m(t− δ): value of m(t) of last sample. The algorithm of (SFAQM) will be depicted as in Figure 1. Int. J. Anal. Appl. 16 (5) (2018) 659 Misra et al. [12], [20] and also Low et al. [19] provided the fluid-flow models to demonstrate the TCP and queue dynamics. Xu and Sun [19] claimed that their model has the same value dropping probability at the operating point. The following model as we mentioned before this is a network with one bottleneck link, with only one TCP flow from each end host.   Ẇi(t) = Wi(t−R) R (1 −p(t−R)) 1 Wi(t) − Wi(t−R) R Wi(t) 2 p(t−R), q̇(t) = −C + ∑N i=1 Wi(t) R , (2.16) where R, Wi(t), p(t) are like pervious section. Using (2.1) in this model, (2.16) changed to:   V̇ (t) = V (t−R) R2V (t) (1 −p(t−R)) − 1 2 V (t−R)V (t)p(t−R), q̇(t) = −C + NV (t). (2.17) The operating point will be derived by V̇ (t) = 0 and q̇(t) = 0 and considering that (V,q) as a state and p as the input, therefore, V̇ (t) = 0 =⇒ p0 = 1 V 20 R 2 2 + 1 , (2.18) q̇(t) = 0 =⇒ V0 = C N . (2.19) Now equation (2.12) is concluded as follows by substituting equation (2.11), (2.13) and (2.14): p(t) = [(1 −ω2)λ(t) + ω2λref (t + τ)]2 + Nδλ(t) R2 −λ(t)λref (t + τ)( [(1−ω2)λ(t)+ω2λref(t+τ)]2 2N + N R2 ) λ(t)δ . (2.20) Then the packet dropping probability will be as follows when the system is at the steady state, and the queue occupation, arrival rate and desired arrival rate are stabilized at qref and C (link capacity), respectively: p0 = 1 C2R2 2N2 + 1 . (2.21) As we mentioned this value is like the same value as Low et al. [19] calculated in their model. For linearizing the above equation the right hand side of equation (2.17) can be written as follows:   f(V,Vτ,pτ ) = Vτ R2V (1 −pτ ) − 12VτV pτ, g(V ) = −C + NV, h(q̇,q) = [(1−ω2)λ+ω2λref]2+Nδλ R2 −λλref( [(1−ω2)λ+ω2λref ] 2 2N +N/R2 ) λδ , (2.22) Int. J. Anal. Appl. 16 (5) (2018) 660 where Vτ (t) = V (t− τ) and λ, λref can be shown as:   λ(t) = ∑N i=1 Vi(t) = q̇(t) + C, λref = C + (qref−q(t)) α . (2.23) Deriving partial derivatives at the operating point in (2.23) we have: ∂f ∂V = − V0(1 −p0) R2V 20 − 1 2 V0p0 = − 1 −p0 R2V0 − 1 2 V0p0 = − N R2C C2R2 2N2 C2R2 2N2 + 1 − 1 2 C N 1 C2R2 2N2 + 1 = − C N 1 C2R2 2N2 + 1 , (2.24) ∂f ∂Vτ = (1 −p0) R2V0 − 1 2 V0p0 = N R2C C2R2 2N2 C2R2 2N2 + 1 − 1 2 C N 1 C2R2 2N2 + 1 = 0, (2.25) ∂f ∂pτ = − 1 2 V 2 = − C2 2N2 , (2.26) ∂g ∂V = N, (2.27) ∂h ∂q̇ = ∂h ∂λ ∂λ ∂q̇ = ∂h ∂λ , (2.28) ∂h ∂q = ∂h ∂λref ∂λref ∂q = − 1 α ∂h ∂λref . (2.29) In order to obtain the two last equations, let a and b be the numerator and denominator of h, respectively. So,   ∂a ∂λ = 2(1 −ω2)C + NδR2 −C, ∂b ∂λ = (3C2 − 2ω22C2) δ 2N + Nδ R2 , ∂a ∂λref = 2ω2C −C, ∂b ∂λref = ω2C 2 N . (2.30) Now we have:   ∂h ∂λ = ∂a ∂λ b−∂b ∂λ a b2 , ∂h ∂λref = ∂a ∂λref b− ∂b ∂λref a b2 . (2.31) Int. J. Anal. Appl. 16 (5) (2018) 661 Thus the linearizing form is as follows:   δV̇ (t) = − C N ( C2R2 2N2 +1 )δV (t) − C2 2N2 δp(t−R), δq̇(t) = NδV (t), δp(t) = ∂p ∂λ δq̇(t) − 1 α ∂p∂λrefδq(t). (2.32) 3. Designing Control model of TCP/AQM Model (2.22) can be formulated as:  ẋp = Apx(t) + Bpu(t−R) + Bdu(t),yp = cpx(t), (3.1) in which x(t) = [δV (t) δq(t)], δp(t) = u(t), y(t) = δq(t) and Ap =   − CN(C2R22N2 +1) 0 N 1 α D2 D1   , Bp =   C22N2 0   , Bd =   0 1 D1   , where D1 = ∂p ∂λ = ∂h ∂λ and D2 = ∂p ∂λref = ∂h ∂λref and also, δq̇ = ( ∂p ∂λ )−1δp(t) + 1 α ( ∂p ∂λref )−1δq(t) = 1 D1 δp(t) + 1 α D2 D1 δq. We can write the state-space as:   ẋp = Apx(t) + Bpu(t−R), yp = cpx(t), u(t) = Dpẋ(t) + Dcx(t), (3.2) where Dp = [0 uλ], Dc = [0 −1 αuref ] and Ap =   − CN(C2R22N2 +1) 0 N 0   , Bp =   C22N2 0   . Int. J. Anal. Appl. 16 (5) (2018) 662 As we can see the model (3.1) is a state-dependent delay differential equation with delay in control. As- suming this model uses a state feedback controller un(t) = Knxp(t) i.e., un(t) = Kn1δV (t) + Kn2δq(t), this equation will changed to:   ẋp = Apxp(t) + BpKpxp(t−R) + BdKnxp(t),yp = cpxp(t), (3.3)   ẋp = (Ap + BdKn)xp(t) + BpKpxp(t−R),yp = cpxp(t), (3.4) A =   − CN(C2R22N2 +1) 0 N + 1 D1 1 α D2 D1   , Ad =   − C22N2 0   . For proofing the stability of state-dependent system which is describe by (3.4), the following theorems are applied whilst stability conditions will be constructed to insure the stability and give the variables of pervious system. First the Lemma which is proposed by Lin [17] is provided as follows and will be used in the proof of aforementioned theorems. Lemma 3.1. If there is exist arbitrary matrices X11, X12, X13, X22, X23 and X33 such that X =   X11 X12 X13 XT12 X22 X23 XT13 X T 23 X33   ≥ 0, (3.5) then we obtain − ∫ t t−RRT(t) ẋT (s)X33ẋ(s)ds ≤ (3.6) ∫ t t−RRT(t) [xT (t) xT (t−RRT(T)) ẋT (s)]   X11 X12 X13 XT12 X22 X23 XT13 X T 23 0     x(t) x(t−RRT(t)) ẋ(s)  ds. Theorem 3.1. Let the scalars h > 0 and µ < 1, the state-dependent delay system (3.3) is asymptotically stable, if the condition (∇RRT.ẋ(t) = λ(t) < µ < 1) is satisfied for every t and there exist positive-definite Int. J. Anal. Appl. 16 (5) (2018) 663 symmetric matrices P , Q and R, and a semi positive-definite matrix X =   X11 X12 X13 X21 X22 X23 X31 X32 X33   , such that the following Linear matrix Inequality(LMI) holds Ap =   E11 E12 hA TR ∗ E22 hATd R ∗ ∗ −hR   < 0, R−X33 ≥ 0, where E11 = A TP + PA + Q + (1 −µ)(X13 + XT13 + hX11), E12 = PAd + (1 −µ)(−X13 + XT23 + hX12), E22 = (1 −µ)(−Q−X23 −XT23 + hX22). Lyapunov stability theory are concept that applied to investigate the ability of these theorems. Shevitz and Paden were proceeded the lyapunov stability for nonsmooth systems [22] and also, Zhang et al. [28] uti- lized feedback control system and provided the stability for networked control systems where their approach is used to establish the stability . Also, Lyapunov-Krasovskii functional and schur complement are using to proof of theorem. Proof: The Lyapunov-Krasovskii that we use here, is as follows: V (xt) = x T (t)Px(t) + ∫ t t−RRT x T (s)Qx(s)ds + ∫ 0 −RRT ∫ t t+θ ẋT (s)Rẋ(s)dsdθ. (3.7) Obviously, this Lyapunov functional candidate is positive definite. Now we calculate the derivative of this functional: V̇ (xt) = ẋ T (t)Px(t) + xTPẋ(t) + xT (t)Qx(t) − (1 − ˙RRT(x))Qx(t−RRT(x)) +RRT(x)ẋT (t)Rẋ(t) − (1 − ˙RRT(x)) ∫ t t−RRT(x) ẋT (s)Rẋ(s)ds. (3.8) Int. J. Anal. Appl. 16 (5) (2018) 664 By substituting (3.7) to (3.8) yields V̇ (xt) = x T (t)[ATP + PA + Q]x(t) + xT (t)PAdx(t−RRT(x)) + (1 −λ(t))xT (t−RRT(x))Qx(t−RRT(x)) + RRT(x)ẋT (t)Rẋ(t) − (1 − ˙RRT(x)) ∫ t t−RRT(x) ẋT (s)(R−X33)ẋ(s)ds − (1 − ˙RRT(x)) ∫ t t−RRT(x) ẋT (s)X33ẋ(s)ds. (3.9) The Leibniz-Newton formula used here and the assumption on ˙RRT(x) < µ and RRT(x) < h. So we have −(1 −µ) ∫ t t−RRT(x) ẋT (s)X33ẋ(s)ds ≤ (1 −µ) ∫ t t−RRT(x) [xT xT (t−RRT(x)) ẋ(s)]   X11 X12 X13 X21 X22 X23 X31 X32 X33     xT xT (t−RRT(x)) ẋ(s)   ≤ (1 −µ)hxT (t)X11x(t) + xThX12x(t−RRT(x)) + xTX13 ∫ t t−RRT(x) ẋ(s)ds + hxT (t−RRT(x))XT12x(t) + hx T (t−RRT(x))X22x(t−RRT(x)) + xT (t−RRT(x))X23 ∫ t t−RRT(x) ẋ(s)ds + XT13x(t) ∫ t t−RRT(x) ẋ(s)ds + XT23x(t−RRT(x)) ∫ t t−RRT(x) ẋ(s)ds = (1 −µ)[xT (t)hX11x(t) + xT (t)hX12x(t−RRT(x)) + xT (t)X13x(t) −x(t−RRT(x))XT13x(t) + x T (t−RRT(x))hXT12x(t) + xT (t−RRT(x))hX22x(t−RRT(x)) + xT (t−RRT(x))X23x(t) −xT (t−RRT(x))X23x(t−RRT(x)) + xT (t)XT13x(t) −x T (t)XT13x(t−RRT(x)) + x(t)X T 23x(t−RRT(x)) −x(t−RRT(x))X T 23x(t−RRT(x))]. By substituting in (3.9), following equation will be concluded: V̇ (xt) < ξ T (t)Zξ(t) − (1 −µ) ∫ t t−RRT(x) ẋ T (s)(R−X33)ẋ(s)ds, (3.10) where ξT (t) = [xT xT (t−RRT(x))] and   z11 z12 0 z22   , Int. J. Anal. Appl. 16 (5) (2018) 665 z11 = A TP + PA + Q + (1 −µ)(X13 + XT13 + hX11) + hA TRA, z12 = PAd + (1 −µ)(−X13 + XT23 + hX12) + hA TRAd, z22 = (1 −µ)(−Q−X23 −XT23 + hX22) + hA T d RAd. In order to achieve to negative derivative of Lyapunov functional, we use the schur complement, (i.e. Z/z11 = z22 −z12z−111 < 0) and LMI conditionals. So the asymptotically stability will be derived. Using the proof of this theorem and substituting AQM model on stability condition yields: V̇ (x) = xT (t) [ − CN(C2R22N2 +1) 0 (N + 1 D1 )Kn 1 α D2 D1   T P + P   − CN(C2R22N2 +1) 0 (N + 1 D1 )Kn 1 α D2 D1   + Q]x(t) + xTP   C22N2 Kp 0  x(t−RRT(x)) + xT (t−RRT(x))[ C2 2N2 Kp 0 ] Qx(t−RRT(x)) × RRT(x)ẋT (t)Rẋ(t) − (1 − ˙RRT(x)) ∫ t t−RRT(x) ẋT (s)(R−X33)ẋ(s)ds − (1 − ˙RRT(x)) ∫ t t−RRT(x) ẋT (s)X33x(s)ds. So the stability of Lyapunov functional, we obtain: V̇ (x) < [xT (t) xT (t−RRT(x))] ×   M N 0 F  × [xT xT (t−RRT(x))]T −(1 −µ) ∫ t t−RRT(x) ẋT (s)(R−X33)ẋ(s)ds < 0,   − CN(C2R22N2 +1) 0 (N + 1 D1 )Kn 1 α D2 D1   T P + P   − CN(C2R22N2 +1) 0 (N + 1 D1 )Kn 1 α D2 D1   + (1 −µ)(X13 + XT13 + hX11) +h   − CN(C2R22N2 +1) 0 (N + 1 D1 )Kn 1 α D2 D1   T R   − CN(C2R22N2 +1) 0 (N + 1 D1 )Kn 1 α D2 D1   < 0, where, M = ATP + PA + (1 −µ)(X13 + XT13 + hX11) + hA TRA, N = PAd + (1 −µ)(X13 + XT23 + hX12) + hARAd, F = (1 −µ)(X23 + XT23 + hX22) + hA T d RAd. Int. J. Anal. Appl. 16 (5) (2018) 666 Simple calculation leads the following equation: (N + 1 D1 )KnP21 + 2( 1 α D2 D1 P22) + h( 1 α D2 D1 )2R22 < 2( C N(C 2R2 2N + 1) )P11 −h( C N(C 2R2 2N + 1) )2R11. (3.11) Theorem 3.2. For any given h and µ < 1, the state dependent delay is asymptotically stabilized via the state feedback controller u(t) = K̃x(t), if the condition (∇RRT.ẋ(t) = λ(t) < µ < 1) satisfies for every t and there exists positive definite symmetric matrices W , U, G and a matrix Y with appropriate dimensions and a semi-positive definite matrix, T =   T11 T12 T13 T21 T22 T23 T31 T32 T33   ≥ 0, such that the following LMI holds, Γ =   Γ11 Γ12 hW   − CN(C2R2/2N2+1) 0 (N + 1 D1 )Kn 1 α D2 D1   ∗ Γ22   C22N2 Kp 0   + hY BTd ∗ ∗ −hG   < 0, and W −T33 ≥ 0, where, Γ11 = W   − CN(C2R2/2N2+1) 0 (N + 1 D1 )Kn 1 α D2 D1  T +   − CN(C2R2/2N2+1) 0 (N + 1 D1 )Kn 1 α D2 D1  W + (1 −µ)(T13 + TT13 + hT11), Γ12 =   C22N2 Kp 0  W +   0 1 D1  Y + (1 −µ)(−T13 + TT23 + hT12), Γ22 = (1 −µ)(−U −T23 −TT23 + hT22), and a stabilizing gain will be driven by:   Kp 0 0 Kn   = Y W−1. Int. J. Anal. Appl. 16 (5) (2018) 667 Proof: Since matrix Ad including controller gains, and its needs to be disassembled, so it should be calculate and we will be factorized matrix Ad as,  C22N2 0 Kp   . Replacing in LMI condition and multiplying to the both sides of that by diagP−1,R−1, defining W = P−1, G = R−1, P−1QP−1 = U, K̃W = Y , P−1XijP −1 = Tij and [R −1 P−1][R −X33]T , P−1 = W −T33 lead to the equation in theorem and the state feedback controller gain then can be found from K̃ = Y W−1 The rate at which the queue length grows when the buffer is nonempty in which Athuraliya et al. [2] men- tioned as mismatch. Also, the AQM method uses state feedback, but Vk −V states might be not available at routers and it makes some difficulties in real network. An approximation uses here to conquer this problem. The control signal for network is un(t) = Knxn(t) which yields: δp(t) = Knxn(t) = [Kn1 Kn2][δq δV ] T , (3.12) (3.13) As [29], the second term of the state vector is investigated: x2 = V −V0 = V − τ0C N = τ0 N ( V N τ0 −C), (3.14) τ0 N [flowrate−C] = τ0 N ∗rate of mismatch. (3.15) This mismatch can be approximated by q̇(t) and it turns to: δp(t) = Kn1.(q −q0) + Kn2. τ0 N q̇(t). (3.16) 4. Numerical Simulation Some Examples of control systems have been simulated here.The parameters ω1, ω2, α and δ will be calculated according to stability condition that we provided in pervious section. [27] determined that qref as reference queue length can be set 500 packets which is the desired value. Also, they set R = 0.1889s and RTPT = 0.1s. Following examples consider the different network condition, actually we alter the network condition . We will see in following examples that all these parameters satisfy the stability condition corre- sponding to these network systems. The network topology that we use here is dumb-bell which the link between router B and router C is the Int. J. Anal. Appl. 16 (5) (2018) 668 Figure 2. The Network Topology [27]. bottleneck link and it’s depicted in figure (2). TCP/Reno is the policy that the sources use. All routers use Droptail except router B where it exploit SFAQM to control queue length. We use the network control system with following features: • Propagation delay is set to 200ms, • Nominal load is 1000 sessions, • Desired queue length is set 100 packets, • Buffer capacity is set 300 packets. In following examples the parameters corresponding to SFAQM policy will be derived according to the stability condition proposed in this paper. It must mentioned that δ is set to 0.00625s. The results explain the effectiveness and good performance of the presented method and also show the these parameters speed of convergence of SFAQM under this stability condition. Example 4.1. Consider the following system which is proposed by Huang and Ngnuang [14] Ap =   0 1 0 0.1   ,Bp =   0 0.1   ,Bd =   0 0.01   . Applying control algorithm proposed in [3] and also considering the theorem (3.2) and relation (3.16) , the controller gains are derived as: up(t) = [−1.4176 − 5.6137]xp(t), un(t) = [0.0002 0.0072]xn(t). The state trajectories of the system which is shown in figure (3), is derived by applying the obtained control into the aforementioned network control system with initial values x0 = [1 1] T . Also the queue length Int. J. Anal. Appl. 16 (5) (2018) 669 Figure 3. The state trajectories of the proposed system . Figure 4. The queue length of network control system. Figure 5. The drop probability as a control law. Figure 6. The state dependent delay. is represented in figure (4). Figure (5), shows the drop probability of system and state dependent delay is depicted in figure (6). The stability of the network control system by proposed method is noticeable by this figures. As shown, the queue length and state dependent delay keep stayed around value. The result is coincide with the result of Azadegan et al. [3], which is represent the affectivity of our proposed method. We must mentioned that we use LMI Toolbox at the MATLAB to obtain this control law. Int. J. Anal. Appl. 16 (5) (2018) 670 Example 4.2. Extending the unstable system where proposed in [25], as follows: ẋ(t) =   −1 0 −0.5 1 −0.5 0 0 0 0.5  x(t) +   0 0 0.1  x(t−R) +   0 0 1   . , and by applying the theorem (3.2), the controlled gain is obtained: up(t) = [0 0 − 0.8762]xp(t), un(t) = [0.0003 0.0018]xn(t), with initial state x0 = [−5 0 5]T . Same as [3] and [25], the state components convergence to zero in faster time as expected. 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