ACTIVE Queue Management (AQM) on Transmission

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1 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS VOL. 17, NO. 2, JUNE 212, 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 previously 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 L 2 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 ACTIVE Queue Management (AQM) on Transmission Control 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 controller 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. Fig. 1. Controller SAT + Closed-Loop System sτ e Plant Antiwindup Within the LMI given controller and compensator propositions, 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 113, Caxias do Sul-RS, Brazil. s: raborsoi@ucs.br, fabender@ucs.br. 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 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 compensators were motivated by the degradation of the transient performance induced by saturation in feedback control systems 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, 1, 8, 2). 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 L 2 -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.

2 48 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 17, NO. 2, JUNE 212 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. presenting 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 L 2 input-to-state stability and internal stability of the closedloop system are therefore carried out from the solution of convex optimization problems. Two optimization criteria are considered for the synthesis: maximization of the L 2 -norm upper bound on the admissible disturbances for which the trajectories are assured bounded and minimization of the L 2 - 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 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 i th 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. C τ = C( τ,,r n ) is the Banach Space of continuous vector functions mapping the interval τ, into R n with the norm φ c = sup φ(t). τ t refers to the Euclidean vector norm. Cτ v is the set def ned by Cτ v = {φ C τ ; φ c < v,v > }. For v R m, sat(v) : R m R m denotes the classical symmetric saturation def ned as sat(v) (i) = sat(v (i) ) = sign(v (i) )min( v (i),u o(i) ), i = 1...m, where u o(i) > denotes the i th magnitude bound. blkdiag{...} is a block diagonal matrix whose diagonal blocks are the ordered elements. 2. PROBLEM STATEMENT Consider the following nonlinear continuous-time delayed system ẋ(t) = Ax(t)+A d x(t τ)+bsat(u(t τ))+b ω ω(t) y(t) = C y x(t) z(t) = C z x(t)+d z u(t) (1) where x(t) R n, u(t) R m, ω(t) R q, y(t) R p, z(t) R l, with τ being constant and the matrices A, A d, B, B ω, C y, C z, and D z of appropriate dimensions. The disturbance vector ω(t) is assumed to be limited in energy, i.e. ω(t) L 2, and for some scalar δ, 1 δ <, the disturbance ω(t) is bounded as follows ω(t) 2 2 = ω(t) ω(t)dt 1 δ Also, the input of the plant is supposed to be limited in amplitude, as def ned in the equation below (2) u o(i) u (i) u o(i), u o(i) >, 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) = A c x c (t)+b c u c (t) (4) y c (t) = C c x c (t)+d c u c (t) where x c (t) R n c, u c(t) R p and y c (t) R m. Matrices A c, B c, C c and D c 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 y a (t) def ned as follows y a (t) = E c ψ(y c (t)) ψ(y c (t)) = sat(y c (t)) y c (t) The controller will end up being described as follows { xc (t) = A c x c (t)+b c u c (t)+y a (t) y c (t) = C c x c (t)+d c u c (t) Def ne now the vector ξ(t) = x(t) x c (t) and the following matrices A Bω A =,R =,B B c C y A c I ω =, nc Ad +BD A d = c C y BC c B,B =,D z = D z, K = D c C y C c,cz = C z +D z C c C y D z C c We can now rewrite the closed-loop system in the form of ξ(t) = Aξ(t)+A d ξ(t τ)+bψ(kξ(t τ)) +RE c ψ(kξ(t))+b ω ω(t) z(t) = C z ξ(t)+d z ψ(kξ(t)) (5) (6) (7)

3 Borsoi et al : Towards Linear Control Approach to AQM in TCP/IP Networks 49 With the initial conditions of the system (7) def ned as φ ξ (θ) = (x(t o +θ) x c (t o +θ) ) φ ξ (θ) = (φ x (θ) φ xc (θ) ) θ τ,,(t o,φ ξ ) R + C v τ 3. MAIN RESULTS 3.1. Local Stabilization Results Theorem 3.1: If there exists symmetric positive def nite matrices Q,Γ R n+nc n+nc, diagonal positive def nite S,S τ R m m, matrices Z R n+nc m, Y,Y τ R m n+nc and scalars α,µ,γ such that the LMIs (8), (9), (1) and (11) are verif ed, there exists E c = ZS 1 such that y a (t) = E c ψ(y c (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) } = φ ξ Cτ v φ 2 c β/(( λ(q 1 )+τ λ(q 1 ΓQ 1 )) with any β so that β µ 1 1 δα ; 2) z(t) 2 2 γv()+γ1 α ω(t) 2 2 ; 3) when ω(t) =, the closed-loop system origin is locally asymptotically stable, and for all { initial conditions belonging to B(µ 1 ) } = φ ξ Cτ v φ 2 c µ 1 /(( λ(q 1 )+τ λ(q 1 ΓQ 1 )). the corresponding trajectories converge asymptotically to the origin. Q K (i) Q+Y (i) µu 2 >, i = 1,...,m (9) Q K (i) Q+Y τ(i) µu 2 >, i = 1,...,m (1) µ αδ < (11) Proof: Consider the following Liapunov-Krasovskii candidate function, and the auxiliar function proposed V(t) = ξ(t) Pξ(t)+ t t τ ξ(θ) Rξ(θ)dθ J(t) = V(t) 1 α ω(t) ω(t)+ 1 γ z(t) z(t) If J <, we have T J(t)dt = V(T) V() T ω(t) ω(t)dt T z(t) z(t)dt <, T > + 1 γ (12) (13) It follows thatξ(t) Pξ(T) V(T) < V()+ ω 2 L 2 β+ δ 1, T >, the trajectories of the system does not leave the set E(P,µ 1 ) for ω(t) satisfying (2); for T, z 2 L 2 γ ω 2 L 2 +γv(); forω(t) =, by def nition, we have V(t) <. This way, in the following development we obtain conditions that once verif ed ensures J(t) <. We evaluate J(t) over the trajectories of the system (7). Expanding V(t) inside J(t) we obtain J(t) = ξ(t) Pξ(t)+ξ(t) P ξ(t) ξ(t) Rξ(t) +ξ(t τ) Rξ(t τ) 1 α ω(t) ω(t)+ 1 γ z(t) z(t) Now, knowing that y c (t) can be rewritten as Kξ(t), and supposing ξ(t) S(u o ) and ξ(t τ) S τ (u o ), with S(u o ) = { ξ K(i) +G (i) ξ uo(i), i = 1,...,m } S τ (u o ) = { ξ K(i) +G τ(i) ξ uo(i), i = 1,...,m } we have the following result Lemma 3.1: 8 If ξ(t) S(u o ) and ξ(t τ) S τ (u o ) then the following relations ψ(y c (t)) T ( ψ(y c (t)) Gξ(t) ) ψ(y c (t τ)) T τ ( ψ(yc (t τ)) G τ ξ(t τ) ) are verif ed for any diagonal positive def nite matrices T,T τ R m m By this relations, we can write that J(t) ξ(t) Pξ(t)+ξ(t) P ξ(t)+ξ(t τ) Rξ(t τ) ξ(t) Rξ(t) 2ψ(y c (t)) Tψ(y c (t)) 1 α ω(t) ω(t) 2ψ(y c (t τ)) T τ ψ(y c (t τ))+ 1 γ z(t) z(t) +ξ(t τ) G τ T τψ(y c (t τ))+ψ(y c (t)) TGξ(t) +ψ(y c (t τ)) T τ G τ ξ(t τ)+ξ(t) G Tψ(y c (t)) Now, let A P +PA+R A d M = P R E c R P +TG 2T B P T τ G τ 2T τ B ω P 1 α I q and C = C z D z Then, M + C 1 γc < implies that J(t) <. Now, by Schur s complement,m+c 1 γ C < is equivalent tom 1 <, where M 1 is given by A P +PA+R A d P R E cr P +TG 2T B P T τ G τ 2T τ B ω P 1 α I q C z D z γi p M 1 < implies that J <, since ξ(t) S(u o ) and ξ(t τ) S τ (u o ), t. We now show that those suppositions are true if E(P,µ 1 ) S(u o ) S τ (u o ) and φ ξ B(β). If φ ξ B(β), it is true that ( λ(p)+τ λ(r)) φ ξ 2 c β. It follows that λ(p) φ ξ 2 c β, and so sup ξ(θ) Pξ(θ) sup λ(p)ξ(θ) ξ(θ) β θ τ, θ τ,) We have that ξ(t) E(P,µ 1 ), t τ,. Hence, if E(P,µ 1 ) S(u o ) S τ (u o ) it follows that ψ(y c (t)) T ( ψ(y c (t)) Gξ(t) ) < and ψ(y c (t τ)) T τ ( ψ(yc (t τ)) G τ ξ(t τ) ) < for t. Then, if E(P,µ 1 ) S(u o ) S τ (u o ) and M 1 < we have that J(t) <.

4 5 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 17, NO. 2, JUNE 212 QA +AQ+Γ QA d Γ Z R +Y 2S S τ B Y τ 2S τ αb ω αi q C z Q D z S γi p < (8) Now pre and post multiplying the left and right hand sides of M 1 by blkdiag{q,q,s,s τ,αi q,i p } with Q = P 1, S = T 1, S τ = Tτ 1 and applying the following variable changes QRQ = Γ, GQ = Y, G τ Q = Y τ, E c S = Z we obtain the matrix inequality stated in (8). By the LMIs in (8), (9) and (1), we can assure the system s trajectories are contained inside the ellipsoid E(P, β), t τ, once E(P,β) S(u o ) S τ (u o ). This is verif ed by the following conditions P K (i) +G (i) µu 2 >,i = 1,...,m P K (i) +G τ(i) µu 2 >,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, G τ Q = Y τ will result in the ellipsoidal inclusion LMIs (9) and (1) 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 L 2 stable for any ω(t) such that ω(t) 2 2 L 2 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 R n+nc n+nc, diagonal positive def nite S,S τ R m m, matrices Z R (n+nc) m and scalars α,µ,γ such that the LMI (14) is verif ed, there exists E c = ZS 1 so that y a (t) = E c ψ(y c (t)) is a static antiwindup compensator that assures that 1) when ω(t), the trajectories of the closed-loop system remains limited for all φ ξ (θ) Cτ v at any initial conditions; 2) z(t) 2 2 γv()+γ1 α ω(t) 2 2 ; 3) if ω(t) =, t t 1,ξ(t) converges asymptotically to the origins. Proof: Let G = K and G τ = K. It follows that the sector condition is verif ed for all ξ(t) R n+nc. In this case, we will have Y = KQ, Y τ = KQ. The remaining of the proof mimics that of Theorem OPTIMIZATION PROBLEMS 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 L 2 -gain upper bound. On these problems, for the sake of simplicity, the inicial condition is assumed to be null (φ ξ (θ) =, t τ,). Also, for the optimization problem (17) it is assumed an a priori value of disturbance 1 δ. In both cases, as β = we thus have µ = δ Maximization of the disturbance tolerance The idea is to maximize the L 2 norm bound on the disturbance 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 < 1.2 µ and (8) (11) Note that since the initial condition is null we have β =, then αµ 1 = 1 δ Maximization of the disturbance attenuation For a non-null bound on the L 2 norm of the admissible disturbances (given by αµ 1 = 1 δ ), the idea is to minimize the upper bound for the L 2 gain 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 < 1.2 γ and (8) (11) 5. NUMERICAL EXAMPLES In this section, we illustrate the Corollary 3.1. Example 1: Consider the system (1) given by the following matrices A =.5, A d =.49975, B = 1, B ω = 1, C y = 1, C z = 1, D z =

5 Borsoi et al : Towards Linear Control Approach to AQM in TCP/IP Networks 51 QA +AQ+Γ QA d Γ Z R KQ 2S S τ B KQ 2S τ αb ω αi q C z Q D z S γi p < (14) And the following output stabilizing controller (4) given by the following matrices A c =.2, B c =.2, C c =.2, D c =.2 The control amplitude is bounded by u o = 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 L 2 disturbance { ω, t t ω(t) =, t t with ω = 12 and t = 1 and a delay τ = 1. Solving the optimization problem 4.2, we obtain the following compensator matrix E c =.7996 Figure 2 depicts the response of the closed-loop system with the obtained antiwindup compensator from (3.1). The controller is given by.1.2 A c = ; B.1 c =.2 C c = ; D c =.2 Applying the results presented in Corollary 3.1, the following compensator matrix was obtained E c = Consider the following L 2 disturbance for the simulation { ω, t t ω(t) =, t t with ω = 1 and t = 1 and a delay τ =.18. Figure 3 depicts the queue lenght and discard probability of the closed-loop system with the obtained antiwindup compensator from (3.1). Plant Output y(t) with AW without AW Variation of queue over average value without AW with AW Plant Input u(t) Time s Variation of discard probability over average value Fig. 2. Global stabilizing results, plant input and output Time s Example 2: Consider the linearized TCP/IP router queue model given in 13. The state variables represent the congestion 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 = 2, τ =.18, C = 22, p =.512 and q = 175. We set u o = Bellow is given the corresponding plant model A = ; B = A d = ; B ω = 1 C y = C z = 1 ; u o =.4898; τ =.18 ; D z = Fig. 3. TCP/IP plant input and output 6. CONCLUSIONS In this work we have presented a methodology for synthesizing 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.

6 52 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 17, NO. 2, JUNE 212 REFERENCES 1 K. J. Astrm and L. Rundqwist. Integrator windup and how to avoid it. In American Control Conference, pages , Pittsburgh, PA, F. A. Bender, J. M. Gomes da Silva Jr. and S. Tarburiech. A convex framework for the design of dynamic anti-windup for state-delayed systems. In American Control Conference, Baltimore, USA, Y. Cao, Z. Wang, J. Tang. Analisys and Anti-Windup Design for Time-Delay Systems Subject to Input Saturation. In IEEE international Conference on Mechatronics ans Automation, Harbin, CN, H. A. Fertik and C. W. Ross. Direct digital control algorithm with antiwindup feature. ISA Transactions, 6: , S. Floyd and V. Jacobson. Random Early Detection gateways for congestion avoidance. IEEE/ACM Transactions on Networking, vol. 1, no. 4, I. Ghiggi, F. A. Bender, and J. M. Gomes da Silva Jr. Dynamic nonrational antiwindup for time-delay systems with saturating inputs. In IFAC 28, Seoul, J. M. Gomes da Silva Jr., F. A. Bender, S. Tarbouriech, and J.-M. Biannic. Dynamic antiwindup for state delay systems: an lmi approach. To appear in CDC J. M. Gomes da Silva Jr. and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an LMI-based approach. IEEE Trans. Autom. Contr., 5(1):16 111, J. M. Gomes da Silva Jr., S. Tarbouriech, and G. Garcia. Anti-windup design for time-delay systems subject to input saturation. an LMI based approach. European Journal of Control, 12: , Dec G. Grimm, J. Hatf eld, I. Postlethwaite, A. Teel, M. Turner, and L. Zaccarian. Anti-windup for stable systems with input saturation: an LMIbased synthesis. IEEE Trans. Autom. Contr., 48(9): , C. V. Hollot, V. Misra, D. Towsley, and W.-B. Gong. A control theoretic analysis of RED. In Proceedings of IEEE/INFOCOM, C. V. Hollot, V. Misra, D. Towsley, and W.-B. Gong. On designing improved controllers for AQM routers supporting TCP f ows. In Proceedings of IEEE/INFOCOM, C. V. Hollot, V. Misra, D. Towsley, and W. Gong. Analysis and design of controllers for AQM routers supporting tcp f ows, IEEE Trans. on Automat. Contr., vol. 47, pp , June N. Kapoor, A. R. Teel, and P. Daoutidis. An anti-windup design for linear systems with input saturation. Automatica, 34(5): , M. V. Kothare and M. Morari. Multiplier theory for stability analisys of anti-windup control systems. Automatica, 35: , V. Misra, W.-B. Gong and D. F. Towsley. Fluid-Based analysis of a network of AQM routers suporting TCP f ows with an application to RED. In Proceedings of ACM SIGCOMM, , S-I. Niculescu. Delay Effects on Stability. A Robust Control Approach. Springer-Verlag, Berlin, Germany, J-K. Park, C-H. Choi, and H. Choo. Dynamic anti-windup method for a class of time-delay control systems with input saturation. Int. J. of Rob. and Nonlin. Contr., 1: , J.P. Richard. Time-delay systems: an overview of some recent advances and open problems. Automatica, 39: , C. Roos and J.-M. Biannic. A convex characterization of dynamically constrained anti-windup controllers. Automatica, pages , Jan C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control via LMI optimization. IEEE Trans. Autom. Contr., 42(7): , S. Tarbouriech, J. M. Gomes da Silva Jr., and G. Garcia. Delay dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs. Int. J. of Rob. and Nonlin. Contr., 14: , S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva Jr. Stability analysis and stabilization of systems presenting nested saturations. IEEE Trans. Autom. Contr., 51: , S. Tarburiech and M. C. Turner. Anti-windup design: an overview of some recent advances and open problems. IEE Proceedings - Control Theory and Applications, pages 1 19, A. R. Teel. Anti-windup for exponentially unstable linear systems. Int. J. of Rob. and Nonlin. Contr., 9(1):71 716, L. Zaccarian, D. Nesic, and A.R. Teel. L 2 anti-windup for linear deadtime systems. Syst. & Contr. Lett., 54(12): , 25. Fernando A. Bender was born in Porto alegre, RS, Brazil. He has graduated in Electric Engineering from UFRGS in 2. Attained his MSc. in Electric Engineering in 26 and his PhD in 21. 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. Borsoi was 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.