A Dynamic Anti-windup AQM for Congestion Control in Internet

A Dynamic An i-windup AQM o Conges ion Con ol in In e ne Nabil EL FEZAZI1, Sadek BELAMFEDEL ALAOUI1, Fa ima EL HAOUSSI1,2, El Houssaine TISSIR1and Te esa ALVAREZ3 Abs ac — This pape add esses he design o an i-windup gains o ob aining s abili y o linea sys ems wi h disc e e ime a ying delay and sa u a ing inpu s delay. Conside ing ha a dynamic ou pu eedback has been designed o s abilize he AQM sys em (wi hou sa u a ion), a me hod is p oposed o designing an an i-windup gain which ela es he sa u a ion o he queue when conges ion, which is ine i able in ne wo ks. I is shown ha he closed-loop sys em ob ained om he con olle plus he an i-windup gain can be modeled by a linea sys em wi h a dead zone nonlinea i y. A modi ied sec o condi ion is hen used o ob ain s abili y condi ions based on Lyapuno unc ions. Di e en ly om p e ious wo ks hese condi ions a e di ec ly in linea ma ix inequali y o m. Resul ing in a new AQM which will be simula ed in MATLAB and compa ed o RED. I. INTRODUCTION Ac i e queue managemen (AQM) is an e ec i e solu ion o he conges ion con ol p oblem. I can achie e high qual- i y o se ice (QoS) by educing he packe d opping p ob- abili y and imp o ing ne wo k u iliza ion. I is implemen ed in ou e o in o m TCP sende s abou a cu en conges ion. Based on his in o ma ion sende s adap s hei sending a e o he s a e o ne wo k. A lo o Ac i e queue managemen a e de eloped in he li e a u e, among hem, Random Ea ly De ec ion (RED): moni o s he a e age queue size and d ops (o ma ks when used in conjunc ion wi h ECN) packe s based on s a is ical p obabili ies, and PI con olle (p opo ional- in eg al con olle ) is a special case o he PID con olle in which he de i a i e (D) o he e o is no used [18], [1], [2]. These con olle s a e widely used in p ac ice, because hei implemen a ion is simple. On he o he hand, many au oma ic con olle s use a i icial in elligence ( uzzy logic, neu al ne wo ks, ...) [3], [6], [11], [17], [18]. These con olle s gi e e y good esul s, bu he implemen a ion o hei algo i hms equi es he p esence o an expe . No e ha he wo ks abo e do no conside plan inpu sa u a ion. Since he con ol ac ion in AQM is de e mining he disca ing p obabili y, which is clea ly a eal numbe bounded be ween [0, 1], you canno ha e a ealis ic AQM ha dis ega ds inpu sa u a ion. In o de o mi iga e he sa u a ion e ec on s abili y o sys- ems, an an i-windup syn hesis o s a e-delayed sys ems has been add essed in [5], [7], [8], [9], [10], [16], [19]. I mus 1Sidi Mohammed Ben Abdellah Uni e si y, Depa men o Physics, Facul y o Sciences Dha El Meh az, LESSI, BP 1796, Fes-A las, Mo occo. [email p o ec ed], [email p o ec ed], [email p o ec ed] 2Polydisciplina y Facul y, BP 300, Selouane 62700, Nado , Mo occo. elhaous [email p o ec ed] 3Valladolid Uni e si y, Depa men o Sys ems Enginee ing and Au oma- ion, 47005, Valladolid, Spain. [email p o ec ed] be no ed ha hese wo ks neglec he sys em disc e izing which is e y impo an o s udy he eal sys em. Thus, we ex end he ou pu dynamic used in [9] and [10] o his wo k. The dynamics o his con olle has been chosen so ha he closed loop sys em is s able. The an i-windup compensa o i sel emi s wo signals, one which is ed di ec ly in o he cons ained con ol signal and one which may be used o d i e he con olle s a e equa ion di ec ly. Vi ually all an i- windup compensa o s which a e p esen in he li e a u e can be ep esen ed in he o m o [16] and he an i-windup compensa o discussed he e will be he same ype. The objec i e o his pape is o design a con olle which is capable o achie ing he queue size and gua an eeing he s abili y o sa u a ed disc e e TCP/AQM sys ems wi h bo h link capaci y dis u bances. Fo his eason, dynamic an i- windup AQM con ol is e y simple o implemen and gi e good esul s. To explain he con olle design, we o ganized his pape as ollows: a p oblem o mula ion unde s udy and use ul lemmas a e p esen ed in Sec ion II. We announce a main heo em and his p oo , an op imiza ion p oblem and an impo an implemen a ion o ou AQM a e discussed in Sec ion III. Finally, a nume ical example is included o illus a e he de eloped esul s. II. PROBLEM FORMULATION AND PRELIMINARIES Ou s udy will ocus on he sha ing o a communica ion link be ween mul iple ansmi e s a emo e loca ions. We conside a single bo leneck ou e unning TCP lows as illus a ed in he ollowing igu e Fig. 1. Simula ion o ne wo k opology The model o TCP beha io ela ing he a e age alue o key ne wo k a iables is desc ibed by he ollowing coupled equa ions [13], [14] ˙ W( )= 1 RTT( )−W( )W( −RTT( )) 2RTT( −RT T ( )) p( −RTT( )) ˙q( )=−C( )+ N( ) RTT( )W( )(1) 978-1-5090-4320-0/16/$31.00 ©2016 IEEE whe e W( )is he a e age TCP window size (packe s); q( )is he a e age queue leng h (packe s); RTT( )is he ound ip ime =q( ) C( )+Tp(secs); Cis he link capaci y (packe s/secs); Tpis he p opaga ion delay (secs); Nis he numbe o sessions; p∈[01]is he p obabili y o packe ma king/d opping. The window size and he queue leng h a e posi i e and bounded; i.e. W∈[0Wmax]and q∈[0qmax].Fo a gi en iple o ne wo k pa ame e s (N,C0,Tp), any iple ℑ=(W0,q0,p0),le Ξ={ℑ:W0=RTTC0 N,q0=C0(RT T −Tp),p0=2 W2 0} be a possible ope a ing poin . Now de ine δ ℭ=ℭ−ℭ0wi h ℭ=W,q,p,C. Then, we can ob ain he linea ized e sion o (1) as ollows δ ˙ W( )= −N RTT2C0( δ W( )+ δ W( −RT T ( ))) −1 RTT2C0( δ q( )+ δ q( −RT T ( ))) −RTTC2 0 2N2 δ p( −RT T ( )) +RTT −Tp RTT2C0( δ C( )+ δ C( −RT T ( ))) δ ˙q( )= N RTT δ W( )−1 RTT δ q( )−Tp RTT δ C( ) RTT( )= δ q( ) C0 +RTT (2) Thus, ew i ing (2) in s a e space o m yields ˙x( )=Ax( )+A τ x( − τ ( )) +Bu( − τ ( )) +Bww( ) y( )=Cyx( ) z( )=Czx( )(3) in which x( )=[ δ W( ) δ q( )],w( )=[ δ C( ) δ C( −RT T ( )) ], u( )= δ p( ),y( )= δ q( ),z( )=RTT( )−RTT, A=[−N RTT2C0−1 RTT2C0 N RTT −1 RTT ],Ad=[−N RTT2C0−1 RTT2C0 00 ], B=[−RTTC2 0 2N2 0],Bw=[RTT−Tp RTT2C0 RTT−Tp RTT2C0 −Tp RTT 0], Cy=[01 ],Cz=[01 C0]. The disc e iza ion o he sys em (3) gi es x(k+1)=Adx(k)+A τ dx(k−d(k))+ Bdu(k−d(k)) +Bwdw(k) y(k)=Cydx(k) z(k)=Czdx(k)(4) whe e Ad=eAT ,A τ d=eA τ T,Cyd=Cy,Czd=Cz,Bd= ∫T 0eA τ (T−s)Bds,Bwd=eBwTand d(k)is a posi i e in ege ep esen ing he ime delay o he sys em ha we assume o be ime dependen and o sa is y he ollowing dm≤d(k)≤dM(5) whe e dmand dMa e known posi i e and ini e in ege s. Then, we ha e Ad=[1−e−T RTT 2C0e−NT RTT 2C0 e−T RTT λ ],A τ d=[e−T RTT λ 00 ], Bd=[1−e−RTTC0T N2 0],Bwd=[1−e−TpT RTT β 00 ] whe e λ =NC0RT T RTT2C0−2N(e−2NT RTT 2C0−e−T RTT )e−NT RTT 2C0and β = e−(RTT −Tp)T RTT 2C0. Thus, he dynamic ou pu s abilizing con olle is conside ed as xc(k+1)=Acxc(k)+Bcy(k) yc(k)=Ccxc(k)+Dcy(k)(6) The in e connec ion o his con olle wi h (4) is gi en by x(k+1)=Adx(k)+A τ dx(k−d(k))+ Bdsa (yc(k−d(k))) +Bwdw(k) y(k)=Cydx(k) z(k)=Czdx(k) xc(k+1)=Acxc(k)+Bcy(k)−Ec ψ (yc(k−d(k))) yc(k)=Ccxc(k)+Dcy(k)(7) The e m Ec ψ (yc(k−d(k))) is injec ed o mi iga e he e ec o windup caused by sa u a ion and ψ (yc(k−d(k))) = yc(k−d(k))−sa (yc(k−d(k))) (8) No e ha , ψ (yc(k−d(k))) co esponds o a decen alized dead-zone nonlinea i y. In his case, he augmen ed sys em can be ep esen ed as ollows ξ (k+1)=𝔸 ξ (k)+𝔸d ξ (k−d(k))−(𝔹+ℝEc) × ψ (𝕂 ξ (k−d(k)))+ 𝔹ww(k) z(k)=ℂz ξ (k)(9) and he augmen ed s a e and ma ices a e gi en by ξ (k)=[x(k) xc(k)],𝔸=[Ad0 BcCydAc],𝔹=[Bd 0], 𝔸d=[A τ d+BdDcCydBdCc 00 ],𝔹w=[Bwd 0], ℝ=[0 Inc],𝕂=[DcCydCc],ℂz=[Czd0]. Deno e by 𝔉zw he closed-loop ans e unc ion om w(k) o z(k). The objec i e o he H∞con ol design is o ind a con olle such ha ∥𝔉zw∥2<√ γ (10) and o minimize γ i possible. Clea ly, γ desc ibes a kind o dis u bance ejec ion a io be ween he con olled a iable and he exogenous dis u bance. Fu he mo e, conside a ma ix G∈ℜm×nand de ine he ollowing polyhed al se S={ ξ (k)∈ℜn;∣(𝕂(k)−G(k)) ξ (k)∣≤u0(k)} he ollowing use ul lemmas will be used in his pape Lemma 2.1: [7] I ξ (k)∈S, hen he ollowing ela ion ψ T(𝕂 ξ (k))T[ ψ (𝕂 ξ (k))−G ξ (k)]≤0 is e i ied o any diagonal posi i e ma ix T∈ℜm×m. III. MAIN RESULTS A. S abili y Resul s In conges ion con ol, one impo an p oblem is o ind maximum allowable uppe bound o he ime delay such ha he ne wo k can s ill be s abilized o ind he H∞pe o mance index can s ill be gua an eed. This p oblem can be easily deal based on he ollowing Theo em. Theo em 3.1: I he e exis symme ic posi i e de ini e ma ices ˆ P,ˆ Q,ˆ R, app op ia ely sized ma ices Fc,ˆ T,ˆ G,ˆ Y1,ˆ Y2, ˆ Y3,ˆ Y4,ˆ Y5and posi i e scala γ sa is ying he LMI (11) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Π11 Π12 Π13 Π14 Π15 Π16 Π17 ∗Π22 Π23 Π24 Π25 Π26 0 ∗∗Π33 Π34 Π35 00 ∗∗∗Π44 0Π46 0 ∗∗∗∗Π55 Π56 0 ∗∗∗∗∗Π66 0 ∗∗∗∗∗∗Π77 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ <0 (11) whe e dMm =dM−dmand Π11 =−ˆ P+ˆ Q+dMm ˆ R−ˆ Y1−ˆ YT 1,Π12 =ˆ Y1−ˆ YT 2 Π22 =−ˆ Q+ˆ Y2+ˆ YT 2,Π13 =ˆ Y1−ˆ YT 3,Π23 =ˆ Y2+ˆ YT 3 Π33 =ˆ Y3+ˆ YT 3,Π14 =−ˆ YT 4,Π24 =ˆ GT+ˆ YT 4,Π34 =ˆ YT 4 Π44 =−2ˆ T,Π15 =−ˆ YT 5,Π25 =Π35 =ˆ YT 5,Π55 =−I Π16 =ˆ P𝔸T,Π26 =ˆ P𝔸T d,Π46 =−ˆ T𝔹T−FT cℝT Π56 =𝔹T w,Π66 =−ˆ P,Π17 =ˆ PℂT z,Π77 =− γ I hen, an an i-windup compensa ion Ec=Fcˆ T−1exis s, such ha he closed-loop sys em (9) sa is ies (1) asymp o ic s abili y; (2) he pe o mance index (10). P oo 1: To p o e his heo em, le us conside he ol- lowing Lyapuno unc ion V(k)=V1(k)+V2(k)+V3(k) = ξ T(k)P ξ (k)+ k−1 ∑ l=k−d(k) ξ T(l)Q ξ (l) +−dm+1 ∑ l=−dM+2 k−1 ∑ m=k+l−1 ξ T(m)R ξ (m)(12) and le us compu e he di e ence o he Lyapuno unc ion ΔV1(k)= ξ T(k+1)P ξ (k+1)− ξ T(k)P ξ (k)(13) ΔV2(k)= k ∑ l=k+1−d(k+1) ξ T(l)Q ξ (l)− k−1 ∑ l=k−d(k) ξ T(l)Q ξ (l) = ξ T(k)Q ξ (k)− ξ T(k−d(k))Q ξ (k−d(k)) + k−1 ∑ l=k+1−dm ξ T(l)Q ξ (l)− k−1 ∑ l=k+1−d(k) ξ T(l)Q ξ (l) + k−dm ∑ l=k+1−d(k+1) ξ T(l)Q ξ (l)(14) ΔV3(k)= −dm+1 ∑ l=−dM+2[k ∑ m=k+l ξ T(m)R ξ (m) − k−1 ∑ m=k+l−1 ξ T(m)R ξ (m)] =dMm ξ T(k)R ξ (k)− k−dm ∑ l=k+1−dM ξ T(l)R ξ (l)(15) whe e dMm =dM−dm. Since (5) wi h ∀Q<R, one can easily see ha k−1 ∑ l=k+1−dm ξ T(l)Q ξ (l)− k−1 ∑ l=k+1−d(k) ξ T(l)Q ξ (l) ≤ k−1 ∑ l=k+1−dm ξ T(l)Q ξ (l)− k−1 ∑ l=k+1−dm ξ T(l)Q ξ (l)=0 (16) k−dm ∑ l=k+1−d(k+1) ξ T(l)Q ξ (l)− k−dm ∑ l=k+1−dM ξ T(l)R ξ (l) ≤ k−dm ∑ l=k+1−d(k+1) ξ T(l)R ξ (l)− k−dm ∑ l=k+1−dM ξ T(l)R ξ (l) ≤ k−dm ∑ l=k+1−dM ξ T(l)R ξ (l)− k−dm ∑ l=k+1−dM ξ T(l)R ξ (l)=0 (17) F om (13)-(17), i ollows ha ΔV(k)= ξ T(k+1)P ξ (k+1)+ ξ T(k)(−P+Q+dMm ×R) ξ (k)− ξ T(k−d(k))Q ξ (k−d(k)) (18) Using he New on-Leibniz o mula p o ides o any app o- p ia ely dimensioned ma ices Y1,...,5yields [ ξ T(k)Y1+ ξ T(k−d(k))Y2+ k−1 ∑ i=k−d(k) yT(i)Y3 + ψ T(𝕂 ξ (k−d(k)))Y4+wT(k)Y5][− ξ (k) + ξ (k−d(k))+ k−1 ∑ i=k−d(k) y(i)]=0 (19) whe e y(i)= ξ (i+1)− ξ (i). On he o he hand, acco ding o he sys em equa ion (9), we ha e ΔV(k)≤[𝔸 ξ (k)+𝔸d ξ (k−d(k))−(𝔹+ℝEc) × ψ (𝕂 ξ (k−d(k)))+ 𝔹ww(k)]T P[𝔸 ξ (k) +𝔸d ξ (k−d(k))−(𝔹+ℝEc) ψ (𝕂 ξ (k−d(k))) +𝔹ww(k)]+ ξ T(k)[−P+Q+dMmR] ξ (k) − ξ T(k−d(k))Q ξ (k−d(k))+ 2[ ξ T(k)Y1 + ξ T(k−d(k))Y2+ k−1 ∑ i=k−d(k) yT(i)Y3+wT(k)Y5 + ψ T(𝕂 ξ (k−d(k)))Y4][− ξ (k)+ ξ (k−d(k)) + k−1 ∑ i=k−d(k) y(i)]−2 ψ T(𝕂 ξ (k−d(k)))T ×[ ψ (𝕂 ξ (k−d(k)))−G ξ (k−d(k))] −wT(k)w(k)+1 γ zT(k)z(k)(20) whe e ΔV(k)=ΔV(k)−wT(k)w(k)+1 γ zT(k)z(k). By simple manipula ion, (20) can be ew i en as ΔV(k)−wT(k)w(k)+1 γ zT(k)z(k)≤ η T(k)Ψ η (k)(21) whe e Ψ=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Ψ11 Ψ12 Ψ13 Ψ14 Ψ15 ∗Ψ22 Ψ23 Ψ24 Ψ25 ∗∗Ψ33 Ψ34 Ψ35 ∗∗∗Ψ44 Ψ45 ∗∗∗∗Ψ55 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , η T(k)=[ ξ T(k) ξ T(k−d(k)) ∑k−1 i=k−d(k)yT(i) ψ T(𝕂 ξ (k−d(k))) wT(k)] and Ψ11 =𝔸TP𝔸−P+Q+dMmR−Y1−YT 1+1 γ ℂT zℂz Ψ12 =𝔸TP𝔸d+Y1−YT 2,Ψ22 =𝔸T dP𝔸d−Q+Y2+YT 2 Ψ13 =Y1−YT 3,Ψ23 =Y2+YT 3,Ψ33 =Y3+YT 3,Ψ34 =YT 4 Ψ14 =−𝔸TP(𝔹+ℝEc)−YT 4,Ψ15 =𝔸TP𝔹w−YT 5 Ψ24 =−𝔸T dP(𝔹+ℝEc)+GTTT+YT 4,Ψ35 =YT 5 Ψ44 =(𝔹+ℝEc)TP(𝔹+ℝEc)−2T,Ψ25 =𝔸T dP𝔹w+YT 5 Ψ45 =−(𝔹+ℝEc)TP𝔹w,Ψ55 =𝔹T wP𝔹w−I The ma ix Ψcan be ew i en as Ψ=ϒ+Γ=ϒ+LTPL whe e ϒ=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ϒ11 ϒ12 ϒ13 ϒ14 ϒ15 ∗ϒ22 ϒ23 ϒ24 ϒ25 ∗∗ϒ33 ϒ34 ϒ35 ∗∗∗ϒ44 0 ∗∗∗∗ϒ55 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ,L=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝔸T 𝔸T d 0 −(𝔹+ℝEc)T 𝔹T w ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ and ϒ11 =−P+Q+dMmR−Y1−YT 1+1 γ ℂT zℂz ϒ12 =Y1−YT 2,ϒ22 =−Q+Y2+YT 2,ϒ13 =Y1−YT 3 ϒ23 =Y2+YT 3,ϒ33 =Y3+YT 3,ϒ14 =−YT 4 ϒ24 =GTTT+YT 4,ϒ34 =YT 4,ϒ44 =−2T ϒ15 =−YT 5,ϒ25 =YT 5,ϒ35 =YT 5,ϒ55 =−I Then, we ha e ΔV(k)−wT(k)w(k)+1 γ zT(k)z(k)≤ η T( )(ϒ+LTPL) η ( ) I is clea ha i ϒ+LTPL <0 (22) hen ΔV(k)−wT(k)w(k)+1 γ zT(k)z(k)<0 (23) Thus, he ollowing condi ion is ob ained by applying he Schu complemen o (22) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ϒ11 ϒ12 ϒ13 ϒ14 ϒ15 𝔸TP ∗ϒ22 ϒ23 ϒ24 ϒ25 𝔸T dP ∗∗ϒ33 ϒ34 ϒ35 0 ∗∗∗ϒ44 0−(𝔹+ℝEc)TP ∗∗∗∗ϒ55 𝔹T wP ∗∗∗∗∗ −P ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ <0 (24) P e- and pos -mul iplying (24) by Δ=diag{P−1,P−1,P−1, T−1,I,P−1}. Then, applying he Schu complemen and aking he ollowing changes o a iables ˆ P=P−1,ˆ T=T−1,Fc=Ecˆ T,ˆ G=Gˆ P,ˆ Y5=Y5ˆ P ˆ Λ=ˆ PΛˆ Pwi hΛ=Q,R,Y1,Y2,Y3,and ˆ Y4=ˆ TY4ˆ P We ob ain he condi ion (11). Since (11) holds, he condi ion (23) is sa is ied. Now, summing up (23) om 0 o ∞wi h espec o k yields, V(∞)<V(0)+ ∞ ∑ k=0(wT(k)w(k)−1 γ zT(k)z(k))(25) Unde he ze o ini ial condi ion V(0)=0 and by no ing ha V(∞)≥0, we ha e (10) which implies ha sys em (9) has i s es ic ed L2−gain om w(k) o z(k)less han γ .Now aking w(k)=0, i is easy o see ha ΔV(k)<0. The p oo is comple ed. B. Op imiza ion P oblem Ve i ying LMIs o Theo em 3.1 ensu es ha he closed loop sys em (9) p esen s bounded ajec o ies o any ad- missible pe u ba ion. Fo a non-null posi i e bound on he L2−no m o he admissible dis u bances, he idea is o minimize he uppe bound o he L2−gain o w(k) on z(k). Conside ing ha he ini ial condi ion is null, his can be ob ained om he solu ion o he ollowing con ex op imiza ion p oblem min γ sub jec o (11)(26) C. Implemen a ion Cons ain s The con ol signal o he ne wo k is gi en by δ p(k)=u(k)=yc(k)=[𝕂1𝕂2𝕂3] ξ (k)(27) In o de o elax he compu ing esou ces o implemen ou p oposal on a eal ne wo k we use an app oxima ion as ollows δ W(k)=W(k)−W0=RT T N(NW(k) RTT −C0) =RTT N( low a e−C0) =RTT N× a e o misma ch(28) Fu he mo e, as in [4], one can also no ice ha he a e o misma ch is he a e a which he queue leng h g ows when he bu e is nonemp y. The e o e, we can app oxima e i by δ q Twhe e 1 Tis he sampling equency. Hence, (27) becomes δ p(k)=(𝕂1 RTT NT +𝕂2) δ q(k)+𝕂3xc(k) =[0𝕂1RTT NT +𝕂2𝕂3] ξ (k)(29) As poin ed ou be o e, o implemen ou AQM con olle using (11), we i s disc e ize (3). IV. ILLUSTRATIVE EXAMPLE The abo e AQM con olle is ob ained ia Theo em 3.1 using he LMI- oolbox o MATLAB. In o de o demon- s a e he e ec i eness and applicabili y o p oposed design me hodology, some Ma lab simula ions o expe imen s a e p o ided o compa e he p oposed design app oach wi h RED con ol scheme o AQM ou e s. A single bo lenecked ou e unning AQM con olle (29) is conside ed in he simula ions. In addi ion o he TCP lows add essed in he model, we also in oduced FTP lows in o he ou e o gene a e a ealis ic a ic scena io. The ound ip ime is RTT =0.253, he bo leneck link capaci y is C0=3750, he ope a ing poin is q0=175, he numbe o connec ions N=60. F om he ollowing model builde W0=RTTC0 N,p0=2 W2 0 and Tp=RTT −q0 C0we can calcula e he s eady s a e disca d p obabili y, he p opaga ion delay and he ound ip ime, espec i ely. Fo his se up, we assume u0=p0. On he o he hand, bellow he ma ices o ou pu s abilizing con olle a e gi en explici ly Ac=0,Bc=1,Cc=8.4969 ×10−6,Dc=1.6996 ×10−5 The simula ion esul s a e e alua ed acco ding o he dis u - bance signal which is de ined as w( )={10,0≤ ≤1 0, ≥1 Then, in o de o compa e he esul s o he p oposed AQM con olle wi h o he AQM ou e design schemes, we in oduce RED con olle in he simula ions. F om [12] a ans e unc ion model o RED is C(s)= K×L ed s+Kwhe e L ed =Pmax max h −min h . The RED pa ame e s a e chosen as ha K=0.005 and he dynamic ange (min h,max h)a e (150,700)packe s. REDs a e aging weigh α =1.33 ×106and Pmax =0.1. Fig. 2 and 3 depic he queue leng h and disca d p obabili y, espec i ely, o bo h AQM con olle and RED wi h ini ial alues ξ 0=[0.50.5]Tand he ob ained an i-windup com- pensa o om Theo em 3.1, Ec=−2.7461 ×10−5whe e dMm =0.246 and T=0.0001. The esolu ion o he op i- miza ion p oblem (26) le us ob ain √ γ =0.1809. Time [s] 10 20 30 40 50 60 70 80 90 100 174.92 174.94 174.96 174.98 175 175.02 175.04 175.06 Theo em 3.1 RED Fig. 2. Va ia ion o a queue o e a e age alue Time [s] 10 20 30 40 50 60 70 80 90 100 ×10-3 7.998 7.9985 7.999 7.9995 8 8.0005 8.001 8.0015 8.002 8.0025 Theo em 3.1 RED Fig. 3. Va ia ion o disca d p obabili y o e a e age alue I can be seen om Fig. 2 ha all queue leng h we e s abi- lized a a ge alue when ou AQM con olle is used. Also, he p oposed me hod has achie ed supe io pe o mance wi h less p obabili y o packe d op compa ed o RED as shown in Fig. 3. This p o es ha he compensa o used can o ce he sys em o apidly achie e he desi ed e e ence alue. In addi ion, he achie ed con olle ackles one o he g ea weakness o p e ious AQM mechanisms as i does no equi e o be uned o di e en ope a ing condi ions. Speci ying he desi ed pe o mance objec i e h ough he desi ed e e ence alue, he p oposed con olle can also mee he o he pe o mance objec i es. V. CONCLUSION I is clea ha he dynamic an i-windup compensa o e- duces he p obabili y d opping packe . The e o e, i inc eases he h oughpu o he ne wo ks o he use s. A con ol heo y app oach has been success ully de eloped o sol e conges ion p oblem in TCP/IP Rou e s. The me hodology has been alida ed by a nume ical example, showing he imp o emen s wi h espec o p e ious app oaches in he li e a u e. The pape can be imp o ed by using a mo e ealis ic model − o In e ne a ic wi hin a la ge pa ame e se . A ypical a ic mix can be used o he e alua ion. Mo e complex ne wo k opologies can be add. These may include he e e se-dumbbell opology wi h mul iple conges ed ga eways and ealis ic In e ne -like opologies such as powe -laws. ACKNOWLEDGMENT Funded by MiCInn DPI2014-54530-R and FEDER unds. REFERENCES [1] Al a ez, T., Design o PID con olle s o TCP/AQM wi eless ne - wo ks, In P oceedings o he Wo ld Cong ess on Enginee ing, 2, pp. 1-8, 2012. 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