VOL. 17, NO. 2, JUNE 2012, 47-52
Towards Linear Control Approach to AQM in
TCP/IP Networks
Ricardo Augusto BORSOI, and Fernando Augusto BENDER
Abstract—In this paper, we are proposing the use of a static antiwindup gain matrix to improve the performance of a pre-viously designed controller used to AQM in congested TCP/IP routers. Considering a system subject to state and input delays, limited disturbance and saturation, the results are given in the form of LMI sets. Theorical results that ensure the asymptotic and the L2 input-to-state stabilities of the closed-loop system
are presented in local as well as global context. The proposed conditions are cast in convex optimization problems. A numerical example illustrates the application of the methodology.
Index Terms—Saturated systems, AQM, TCP/IP, stability, LMIs, antiwindup.
1. INTRODUCTION
1.1. Networks and AQM
A
CTIVE Queue Management (AQM) on TransmissionControl Protocol/Internet Protocol (TCP/IP) networks is a very important research area in Telecom for its relation with internet traff c congestion and quality of service (QoS) demands of end users and applications. Since [5], when the Random Early Detection (RED) algorithm was proposed, this area has received much attention and research in the community. Recently, control approachs have been applied to TCP/AQM [16], [11]. In [12], a Proportional-Integral con-troller was proposed. Since then, the control techniques usage has grown signif cantly, but very few works have considered the systhesis of a controller given by Linear Matrix Inequalities (LMI) conditions. We propose in this work the systhesis of an antiwindup compensator for the linearized model previously mentioned, as it can be seen in Figure 1, with a delay independent framework. SAT Controller Plant Antiwindup −s e τ + −Fig. 1. Closed-Loop System
Within the LMI given controller and compensator proposi-tions, even less works consider delay independent results. The
R. A. Borsoi and F. A. Bender are with UCS - Center of Exact Sciences and Technology, R. Francisco Getlio Vargas 1130, 95070-560 Caxias do Sul-RS, Brazil. e-mails: [email protected], [email protected].
proposition of an antiwindup by a delay dependent approach can be seen in [3], but it should be noticed that this framework have a great disadvantage on the network application: the uncertainty associated with the instant delay present on the system can lead to instability of the closed-loop system and loss of performance. So, a delay-independent approach can provide a more stable compensator considering the much varying delay present on real networks, as the stability can be assured for any present value.
1.2. Antiwindup
The antiwindup compensation is a well-known and eff cient technique to cope with undesirable effects (on performance and stability) produced by actuator saturation in control loops. The f rst results regarding the design of antiwindup com-pensators were motivated by the degradation of the transient performance induced by saturation in feedback control sys-tems containing integral actions. See for instance [4], [1]. More recently, the study of the antiwindup problem has been considered in a formal context and a large amount of systematic synthesis methods have been proposed (see for instance [14], [25] and the survey [24] for a large overview). In particular, some of these works are based on LMI (or almost LMI) conditions (see among others [15], [10], [8], [20]). The advantage of the LMI-based methods lies on the fact that the antiwindup design can be carried out through convex optimization problems. In this case different optimal synthesis criteria (such as L2-gain attenuation or enlargement of the basin of attraction) can be directly addressed in an optimal way.
Besides the actuator saturation, it is well-known that time delays are present in many control applications and are also source of performance degradation and even instability (see for instance [17], [19] and references therein). However, it appears that most of the antiwindup design methods (as the ones mentioned in the previous paragraph) regards only undelayed systems. The antiwindup compensation for timedelay systems, was addressed, for instance, in [18], [26], [9] and [22]. In [18] and [26] plants subject to input and/or output delays are considered. For this case, it is considered the synthesis of a dynamic antiwindup compensator aiming at minimizing a cost function. The cost function measures the absolute difference between the controller state considering saturation free actuators and the controller state when the plant input saturation is considered. It should however be pointed out that the results apply only to stable open loop systems and that the approach does not consider systems presenting state delays.
In [9] and [22], an LMI approach to synthesize stabilizing static antiwindup has been proposed. Differently from the classical objective of recovering performance, in those works the antiwindup compensation has been used to enlarge the region of attraction of the closed-loop system. In particular, the action of disturbances and closed-loop performance issues were not considered in these works. The dynamic antiwindup synthesis for state delayed systems has been recently addressed in [6] and [7]. The approach in [6] was based on congruence transformations, similar to the ones proposed in [21], allowing only the synthesis of non-rational compensators (i.e. present-ing delayed terms in the dynamics). From a projection Lemma approach, in [7] it is shown that the synthesis of rational compensator can be carried out by true LMI conditions. In [2] the synthesis of non-rational and rational compensators based on congruence transformations was given, but it only considered state delayed systems. It should be noted that all the previous works dont consider both state and input delayed systems with a delay independent framework.
In this work, we address the problem of synthesizing static antiwindup compensators for state and input delayed linear systems. Based on the use of a Liapunov-Krasovskii approach and a generalized sector condition, true LMI conditions for the synthesis of the antiwindup compensator is proposed. Results concerning the guarantee of local (regional) input-to-state as well as asymptotic stability are obtained and from them, the global case is derived as a particular case. The computation of the antiwindup compensator aiming at ensuring both L2 input-to-state stability and internal stability of the closed-loop system are therefore carried out from the solution of convex optimization problems. Two optimization criteria are considered for the synthesis: maximization of the L2-norm upper bound on the admissible disturbances for which the trajectories are assured bounded and minimization of the L2 -gain of the disturbance to the system regulated output.
This paper is organized as follows. In Section 2, the problem treated in this work is formally stated. Section 3 presents Theorem 3.1 for the local stability antiwindup gain computation, as well as the Corollary 3.1 for global stability case and the algebraic development. Section 4 presents some convex optimization problems, based on the statements of Section 3. Numerical examples are presented in Section 5, and some concluding remarks ends this paper in Section 6.
1.3. Notations
For two symmetric matrices, A and B, A > B means that A − B is positive def nite. A′ denotes the transpose of A. A(i) denotes the ith line of matrix A. ⋆ stands for symmetric blocks. I denotes an identity matrix of appropriate order. λ(P ) and λ(P ) denote the minimal and maximal eigenvalues of matrix P , respectively. Cτ = C([−τ, 0], ℜn) is the Banach Space of continuous vector functions mapping the interval [−τ, 0] into ℜn with the norm kφk
c = sup
−τ ≤t≤0kφ(t)k. k · k refers to the Euclidean vector norm. Cv
τ is the set def ned by Cv
τ = {φ ∈ Cτ; kφkc< v, v > 0}. For v ∈ ℜm, sat(v) : ℜm→ ℜmdenotes the classical symmetric saturation def ned as sat(v)(i) = sat(v(i)) = sign(v(i))min(|v(i)|, uo(i)), ∀i =
1 . . . m, where uo(i) > 0 denotes the ith magnitude bound. blkdiag {. . . } is a block diagonal matrix whose diagonal blocks are the ordered elements.
2. PROBLEMSTATEMENT
Consider the following nonlinear continuous-time delayed system ˙x(t) = Ax(t) + Adx(t − τ ) + Bsat(u(t − τ)) + Bωω(t) y(t) = Cyx(t) z(t) = Czx(t) + Dzu(t) (1) where x(t) ∈ ℜn, u(t) ∈ ℜm, ω(t) ∈ ℜq, y(t) ∈ ℜp, z(t) ∈ ℜl, with τ being constant and the matrices A, A
d, B, Bω, Cy, Cz,and Dz of appropriate dimensions.
The disturbance vector ω(t) is assumed to be limited in energy, i.e. ω(t) ∈ L2, and for some scalar δ, 0 ≤ 1δ < ∞, the disturbance ω(t) is bounded as follows
kω(t)k22= Z ∞
0
ω(t)′ω(t)dt ≤1
δ (2)
Also, the input of the plant is supposed to be limited in amplitude, as def ned in the equation below
−uo(i) ≤ u(i) ≤ uo(i), uo(i) > 0, i = 1, . . . , m (3)
And now consider the following dynamic output stabilizing controller, designed without considering the plant limitations in (2) and (3)
˙
xc(t) = Acxc(t) + Bcuc(t)
yc(t) = Ccxc(t) + Dcuc(t) (4) where xc(t) ∈ ℜnc, uc(t) ∈ ℜp and yc(t) ∈ ℜm. Matrices Ac, Bc, Cc and Dc are also of appropriate dimensions. To mitigate the effects of the windup caused by saturation, we add to the state of the previously designed controller an antiwindup signal ya(t) def ned as follows
ya(t) = Ecψ(yc(t))
ψ(yc(t)) = sat(yc(t)) − yc(t) (5) The controller will end up being described as follows
˙
xc(t) = Acxc(t) + Bcuc(t) + ya(t)
yc(t) = Ccxc(t) + Dcuc(t) (6) Def ne now the vector ξ(t) = x(t)′ x
c(t)′ ′ and the following matrices A= A 0 BcCy Ac , R = 0 Inc , Bω= Bω 0 , Ad= Ad+ BDcCy BCc 0 0 , B = B 0 , Dz= Dz, K= DcCy Cc , Cz= Cz+ DzCcCy DzCc We can now rewrite the closed-loop system in the form of
˙ξ(t) = Aξ(t) + Adξ(t − τ ) + Bψ(Kξ(t − τ )) +REcψ(Kξ(t)) + Bωω(t)
z(t) = Czξ(t) + Dzψ(Kξ(t))
With the initial conditions of the system (7) def ned as φξ(θ) = (x(to+ θ)′ xc(to+ θ)′) ′ φξ(θ) = (φx(θ)′ φxc(θ) ′ )′ ∀θ ∈[−τ, 0], (to, φξ) ∈ ℜ+× Cτv 3. MAINRESULTS
3.1. Local Stabilization Results
Theorem 3.1: If there exists symmetric positive def nite
matrices Q, Γ ∈ ℜn+nc×n+nc, diagonal positive def nite
S, Sτ ∈ ℜm×m, matrices Z ∈ ℜn+nc×m, Y, Yτ ∈ ℜm×n+nc and scalars α, µ, γ such that the LMIs (8), (9), (10) and (11) are verif ed, there exists Ec = ZS−1 such that ya(t) = Ecψ(yc(t)) is a static antiwindup compensator that assures that
1) the trajectories of the system (7) are bounded for every initial condition in the ball B(β) = n φξ∈ Cτv kφk 2 c ≤ β/((¯λ(Q−1) + τ ¯λ(Q−1ΓQ−1)) o
with any β so that 0 ≤ β ≤ µ−1− 1 δα; 2) kz(t)k2 2≤ γV (0) + γ 1 αkω(t)k 2 2;
3) when ω(t) = 0, the closed-loop system origin is locally asymptotically stable, and for all initial conditions belonging to B(µ−1) = n φξ∈ Cτv kφk 2 c≤ µ −1/((¯λ(Q−1) + τ ¯λ(Q−1ΓQ−1))o . the corresponding trajectories converge asymptotically to the origin.
Q ⋆
K(i)Q + Y(i) µu2 o(i)
> 0, i = 1, . . . , m (9)
Q ⋆
K(i)Q + Yτ(i) µu2 o(i)
> 0, i = 1, . . . , m (10)
µ − αδ < 0 (11)
Proof: Consider the following Liapunov-Krasovskii
can-didate function, and the auxiliar function proposed V (t) = ξ(t)′P ξ(t) + Rt t−τ ξ(θ)′Rξ(θ)dθ J (t) = V (t) −˙ α1ω(t)′ω(t) + 1 γz(t) ′z(t) (12) If J < 0, we have RT 0 J (t)dt = V (T ) − V (0) − RT 0 ω(t) ′ω(t)dt +γ1RT 0 z(t) ′z(t)dt < 0, ∀T > 0 (13) It follows that ξ(T )′P ξ(T ) ≤ V (T ) < V (0)+||ω||2 L2 ≤ β+
δ−1, ∀T > 0, the trajectories of the system does not leave the set E(P, µ−1) for ω(t) satisfying (2); for T → ∞, ||z||2
L2 ≤
γ ||ω||2L2+γV (0); for ω(t) = 0, by def nition, we have ˙V (t) <
0.
This way, in the following development we obtain condi-tions that once verif ed ensures J (t) < 0. We evaluate J (t) over the trajectories of the system (7). Expanding V (t) inside J (t) we obtain J (t) = ˙ξ(t)′P ξ(t) + ξ(t)′P ˙ξ(t) − ξ(t)′Rξ(t) + ξ(t − τ )′Rξ(t − τ ) − 1 αω(t) ′ω(t) + 1 γz(t) ′z(t)
Now, knowing that yc(t) can be rewritten as Kξ(t), and supposing ξ(t) ∈ S(uo)and ξ(t − τ) ∈ Sτ(uo), with
S(uo) =ξ |K(i)+ G(i) ξ | ≤ uo (i), i = 1, . . . , m Sτ(uo) =ξ |K(i)+ Gτ (i) ξ | ≤ uo(i), i = 1, . . . , m
we have the following result
Lemma 3.1:[8] If ξ(t) ∈ S(uo)and ξ(t−τ) ∈ Sτ(uo)then
the following relations
ψ(yc(t))′T ψ(yc(t)) − Gξ(t) ≤ 0 ψ(yc(t − τ ))′Tτ ψ(yc(t − τ )) − Gτξ(t − τ ) ≤ 0 are verif ed for any diagonal positive def nite matrices T, Tτ ∈ ℜm×m
By this relations, we can write that
J (t) ≤ ˙ξ(t)′P ξ(t) + ξ(t)′P ˙ξ(t) + ξ(t − τ )′Rξ(t − τ ) −ξ(t)′Rξ(t) − 2ψ(y c(t))′T ψ(yc(t)) −α1ω(t)′ω(t) −2ψ(yc(t − τ ))′Tτψ(yc(t − τ )) +γ1z(t)′z(t) +ξ(t − τ )′G′ τTτψ(yc(t − τ )) + ψ(yc(t))′T Gξ(t) +ψ(yc(t − τ ))′TτGτξ(t − τ ) + ξ(t)′G′T ψ(yc(t)) Now, let M= A′P + P A + R ⋆ ⋆ ⋆ ⋆ A′ dP −R ⋆ ⋆ ⋆ E′ cR′P + T G 0 −2T ⋆ ⋆ B′P T τGτ 0 −2Tτ ⋆ B′ ωP 0 0 0 −α1Iq and C= Cz 0 Dz 0 0 Then, M + C′1
γC < 0 implies that J (t) < 0. Now, by Schur’s complement, M+C′1 γC< 0is equivalent to M1< 0, where M1 is given by A′P + P A + R ⋆ ⋆ ⋆ ⋆ ⋆ A′ dP −R ⋆ ⋆ ⋆ ⋆ E′ cR′P + T G 0 −2T ⋆ ⋆ ⋆ B′P TτGτ 0 −2Tτ ⋆ ⋆ B′ ωP 0 0 0 −α1Iq ⋆ Cz 0 Dz 0 0 −γIp
M1< 0implies that J < 0, since ξ(t) ∈ S(uo)and ξ(t − τ ) ∈ Sτ(uo), ∀t ≥ 0. We now show that those suppositions are true if E(P, µ−1) ⊂ S(u
o) ∩ Sτ(uo)and φξ∈ B(β). If φξ ∈ B(β), it is true that (¯λ(P ) + τ ¯λ(R))||φξ||2c≤ β. It follows that ¯λ(P )||φξ||2c ≤ β, and so
sup θ∈[−τ,0] ξ(θ)′P ξ(θ) ≤ sup θ∈[−τ,0) ¯ λ(P )ξ(θ)′ξ(θ) ≤ β We have that ξ(t) ∈ E(P, µ−1), ∀ t ∈ [−τ, 0]. Hence, if E(P, µ−1) ⊂ S(u
o) ∩ Sτ(uo) it follows that ψ(yc(t))′T ψ(yc(t)) − Gξ(t)
< 0 and ψ(yc(t − τ ))′T τ ψ(yc(t − τ )) − Gτξ(t − τ ) < 0 for t ≥ 0. Then, if E(P, µ−1) ⊂ S(u
o) ∩ Sτ(uo) and M1 < 0 we have that J (t) < 0.
QA′+ AQ + Γ ⋆ ⋆ ⋆ ⋆ ⋆ QA′ d −Γ ⋆ ⋆ ⋆ ⋆ Z′R′+ Y 0 −2S ⋆ ⋆ ⋆ SτB′ Yτ 0 −2Sτ ⋆ ⋆ αB′ ω 0 0 0 −αIq ⋆ CzQ 0 DzS 0 0 −γIp < 0 (8)
Now pre and post multiplying the left and right hand sides of M1by blkdiag {Q, Q, S, Sτ, αIq, Ip}with Q = P−1, S = T−1, S
τ = Tτ−1and applying the following variable changes QRQ = Γ, GQ = Y, GτQ = Yτ, EcS = Z we obtain the matrix inequality stated in (8). By the LMIs in (8), (9) and (10), we can assure the system’s trajectories are contained inside the ellipsoid E(P, β), ∀t ≥ −τ, once E(P, β) ⊂ S(uo) ∩ Sτ(uo). This is verif ed by the following conditions
P ⋆
K(i)+ G(i) µu2 o(i)
> 0, i = 1, . . . , m
P ⋆
K(i)+ Gτ(i) µu2o
(i)
> 0, i = 1, . . . , m
These matrices, by being pre and post multiplied by blkdiag {Q, I}, with Q = P−1 and applying the variable changes GQ = Y, GτQ = Yτ will result in the ellipsoidal inclusion LMIs (9) and (10).
3.2. Global Stability Results
When the plant is asymptotically stable, it is possible to seek the assurance of the global stability of the closed-loop system, i.e. the closed-loop system is L2 stable for any ω(t) such that kω(t)k2
2 ∈ L2 and the origin of the system is globally asymptotically stable. The following corollary extends the result of Theorem 3.1 to the global case.
Corolary 3.1: If there exists symmetric positive def nite
matrices Q, Γ ∈ ℜn+nc×n+nc, diagonal positive def nite
S, Sτ ∈ ℜm×m, matrices Z ∈ ℜ(n+nc)×m and scalars α, µ, γ such that the LMI (14) is verif ed, there exists Ec= ZS−1so that ya(t) = Ecψ(yc(t)) is a static antiwindup compensator that assures that
1) when ω(t) 6= 0, the trajectories of the closed-loop system remains limited for all φξ(θ) ∈ Cτv at any initial conditions; 2) kz(t)k2 2≤ γV (0) + γα1kω(t)k 2 2; 3) if ω(t) = 0, ∀t ≥ t1≥ 0, ξ(t)converges asymptotically to the origins.
Proof: Let G = −K and Gτ = −K. It follows that the
sector condition is verif ed for all ξ(t) ∈ ℜn+nc. In this case,
we will have Y = −KQ, Yτ = −KQ. The remaining of the proof mimics that of Theorem 3.1.
4. OPTIMIZATIONPROBLEMS
The LMIs of Theorem 3.1 and Corollary 3.1 ensure that the closed-loop system presents bounded trajectories for any admissible disturbance, provided that the initial conditions
belong to the set B(β). Since the proposed conditions are in the form of LMI sets, they are considered convex optimization problems. We have two optimization problems in the sequel. On the f rst one, we consider the maximization of the bound 1/δ on the admissible disturbance, for which the trajectories remain bounded. Later on, the second problem regards the minimization of an L2-gain upper bound. On these problems, for the sake of simplicity, the inicial condition is assumed to be null (φξ(θ) = 0, ∀t ∈ [−τ, 0]). Also, for the optimization problem (17) it is assumed an a priori value of disturbance 1
δ. In both cases, as β = 0 we thus have µ = δ.
4.1. Maximization of the disturbance tolerance
The idea is to maximize the L2 norm bound on the distur-bance for which it can be ensured that the system trajectories remain bounded. Hence, the maximization of the disturbance tolerance can be achieved as follows
min µ (15)
subject to (8) − (11)
max α (16)
subject to µ < 1.20 · µ and (8) − (11)
Note that since the initial condition is null we have β = 0, then αµ−1=1
δ.
4.2. Maximization of the disturbance attenuation
For a non-null bound on the L2 norm of the admissible disturbances (given by αµ−1 = 1
δ), the idea is to minimize the upper bound for the L2gain of ω(t) on z(t). Considering that the initial condition is null, this can be obtained from the solution of the following convex optimization problems
min γ (17)
subject to (8) − (11)
max α (18)
subject to γ < 1.20 · γ and (8) − (11)
5. NUMERICALEXAMPLES
In this section, we illustrate the Corollary 3.1.
Example 1:Consider the system (1) given by the following
matrices
A = −0.5, Ad= 0.049975, B = 1, Bω= 1, Cy= 1, Cz= 1, Dz= 0
QA′+ AQ + Γ ⋆ ⋆ ⋆ ⋆ ⋆ QA′ d −Γ ⋆ ⋆ ⋆ ⋆ Z′R′− KQ 0 −2S ⋆ ⋆ ⋆ SτB′ −KQ 0 −2Sτ ⋆ ⋆ αB′ ω 0 0 0 −αIq ⋆ CzQ 0 DzS 0 0 −γIp < 0 (14)
And the following output stabilizing controller (4) given by the following matrices
Ac = −0.2, Bc= 0.2, Cc= 0.2, Dc= 0.2
The control amplitude is bounded by uo = 1. Now, the open loop system is stable. Therefore we can apply the global stability results presented in Corollary 3.1. We then solve the optimization problem 4.2. For simulation purposes consider the following L2 disturbance
ω(t) =
¯
ω, 0 ≤ t ≤ ¯t 0, t ≥ ¯t
with ¯ω = 120 and ¯t = 1 and a delay τ = 1. Solving the op-timization problem 4.2, we obtain the following compensator matrix
Ec = 0.7996
Figure 2 depicts the response of the closed-loop system with the obtained antiwindup compensator from (3.1).
0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5
Plant Output − y(t)
0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 Time [s]
Plant Input − u(t)
with AW without AW
Fig. 2. Global stabilizing results, plant input and output
Example 2: Consider the linearized TCP/IP router queue
model given in [13]. The state variables represent the conges-tion window and the queue size, respectively, the disturbance is User Datagram Protocol (UDP) traff c, and the input is the packet discard probability. The setup is N = 20, τ = 0.18, C = 220, p0 = 0.5102 and q0 = 175. We set uo = 0.4898. Bellow is given the corresponding plant model.
A = −2.8058 −0.1403 111.1111 −5.5556 ; B = −10.89 0 Ad= −2.8058 0.1403 0 0 ; Bω= 0 1 ; Dz= 0 Cy= Cz= 0 1 ; uo= 0.4898; τ = 0.18
The controller is given by Ac = −0.1 0 0 −0.1 ; Bc = 0.2 0.2 Cc= 0.0113 0.0113 ; Dc= −0.02
Applying the results presented in Corollary 3.1, the follow-ing compensator matrix was obtained
Ec=
8.0345 3.4670
Consider the following L2 disturbance for the simulation ω(t) =
¯
ω, 0 ≤ t ≤ ¯t 0, t ≥ ¯t
with ¯ω = 1000 and ¯t= 10 and a delay τ = 0.18.
Figure 3 depicts the queue lenght and discard probability of the closed-loop system with the obtained antiwindup com-pensator from (3.1). 0 5 10 15 20 25 30 35 0 50 100 150 200
Variation of queue over average value
without AW with AW 0 5 10 15 20 25 30 35 −0.4 −0.2 0 0.2 0.4 0.6 Time [s]
Variation of discard probability over average value
Fig. 3. TCP/IP plant input and output
6. CONCLUSIONS
In this work we have presented a methodology for synthe-sizing static antiwindup compensators for system subjected to state and input delays and input saturation. The conditions that ensure the existence of a solution are obtained in an LMI form, which allows to formulate the antiwindup synthesis problem directly as a convex optimization problem, avoiding the use of iterative schemes. Finally, numerical examples are employed to demonstrate the effectiveness of our approach.
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Fernando A. Bender was born in Porto alegre,
RS, Brazil. He has graduated in Electric Engineering from UFRGS in 2000. Attained his MSc. in Electric Engineering in 2006 and his PhD in 2010. Currently he is professor and researcher at Universidade de Caxias do Sul, Caxias do Sul, Brazil. His main research interest is linear control of time delayed saturating systems, with applications in Telecom Networks.
Rodrigo A. Borsoiwas born in Farroupilha, Brazil.
Currently, he is an undergraduate student in Control and Automation Engineering at University of Caxias do Sul. His research interests include control under restrictions, adaptive control and control of systems under the presence of delay.
